Apparatus for generating focused electromagnetic radiation

ABSTRACT

An apparatus for generating electromagnetic radiation comprises a polarizable or magnetizable medium. A polarization or magnetisation current can be generated, in a controlled manner, whose distribution pattern has an accelerated motion, so that non-spherically decaying and intense spherically decaying components of electromagnetic radiation can be generated. The coordinated motion of aggregates of charged particles can give rise to extended electric charges and currents. The charged distribution patterns can propagate with a phase speed exceeding the speed of light in vacuo and that, once created, such propagating charged patterns act as sources of electromagnetic fields in precisely the same way as any other moving sources of these fields. That the distribution patterns of these sources travel faster than light is not, of course, in any way incompatible with the requirements of special relativity. The superluminally moving charged pattern is created by the coordinated motion of aggregates of subluminally moving particles.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation of co-pending U.S. application Ser.No. 11/389,183, filed Mar. 27, 2006 and entitled “APPARATUS FORGENERATING FOCUSED ELECTROMAGNETIC RADIATION,” which is a continuationof U.S. application Ser. No. 09/786,507, filed May 1, 2001, which is theU.S. national phase of international application PCT/GB1999/002943,filed 6 Sep. 1999, designating the U.S. and claiming priority from GB9819504.3, filed 7 Sep. 1998, the entire contents of each of which arehereby incorporated by reference.

DETAILED DESCRIPTION

The present invention relates to the generation of electromagneticradiation and, more particularly, to an apparatus and method ofgenerating focused pulses of electromagnetic radiation over a wide rangeof frequencies. More particularly it relates to an apparatus and methodfor generating pulses of non-spherically decaying electromagneticradiation.

The present apparatus and method are based on the emission ofelectromagnetic radiation by rapidly varying polarisation ormagnetisation current distributions rather than by conduction orconvection electric currents. Such currents can have distributionpatterns that move with arbitrary speeds (including speeds exceeding thespeed of light in vacuo), and so can radiate more intensely over a muchwider range of frequencies than their conventional counterparts. Thespectrum of the radiation they generate could extend to frequencies thatare by many orders of magnitude higher than the characteristic frequencyof the fluctuations of the source itself.

Furthermore, intensities of normal emissions decay at a rate of R⁻²,where R is the distance from the source. It has been noted, however,that the intensities of certain pulses of electromagnetic radiation candecay spatially at a lower rate than that predicted by this inversesquare law (see Myers et al., Phys. World, November 1990, p. 39). Thenew solution of Maxwell's equations set out below, for example, predictsthat the electromagnetic radiation emitted from superluminally,circularly moving charged patterns decays at a rate of R⁻¹. Anotherexample is the electromagnetic radiation emitted from superluminally,rectilinearly moving charged patterns which decays at a rate of

$R^{- \frac{2}{3}}.$

This emission process can be exploited, moreover, to generate waveswhich do not form themselves into a focused pulse until they arrive attheir intended destination and which subsequently remain in focus onlyfor an adjustable interval of time.

It will be widely appreciated that being able to employ such emissionsfor signal transmission, amongst other applications, would havesignificant commercial value, given that it would enable the employmentof lower power transmitters and/or larger transmission ranges, the useof signals that cannot be intercepted by third parties, and theexploitation of higher bandwidth. The near-field component of theradiation in question has many features in common with, and so can beused as an alternative to, synchrotron radiation. The present inventionprovides a method and apparatus for generating such emissions.

According to the present invention there is provided an apparatus forgenerating electromagnetic radiation comprising:

a polarizable or magnetizable medium; and

means of generating, in a controlled manner, a polarisation ormagnetisation current with a rapidly moving, accelerating distributionpattern such that the moving source in question generateselectromagnetic radiation.

The speed of the moving distribution pattern may be superluminal so thatthe apparatus generates both a non-spherically decaying component and anintense spherically decaying component of electromagnetic radiation.

The apparatus may comprise a dielectric substrate, a plurality ofelectrodes positioned adjacent to the substrate, and the means forapplying a voltage to the electrodes sequentially at a rate sufficientto induce a polarised region in the substrate which moves along thesubstrate with a speed exceeding the speed of light. The dielectricsubstrate may have either a rectilinear or a circular shape.

The wavelength of the generated electromagnetic radiation may be in anyrange from the radio to a minimum determined only by the lower limit tothe acceleration of the source (potentially optical, ultraviolet or evenx-ray).

Examples of the present invention will now be described with referenceto the accompanying drawings, in which:

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram showing the wave fronts of the electromagneticemission from a particular volume element (source point) S within thecircularly moving polarised region of the polarizable medium of thepresent invention;

FIG. 2 is a graph showing the value of a function representing theemission time versus the retarded position for differing source pointsa, b, c within the polarizable medium in question;

FIG. 3A is a perspective view of the envelope of the wave fronts shownin FIG. 1 showing the radiation pattern of a single volume element ofthe source;

FIG. 3B is a representative three dimensional plot of the radiationpattern of the entire source of FIG. 7B at a frequency of 2.4 GHz and aphase difference between adjacent electrodes of 15 degrees;

FIG. 3C is a representative three dimensional plot of the radiationpattern of the entire source of FIG. 7B at a frequency of 2.4 GHz and aphase difference between adjacent electrodes of 5 degrees;

FIG. 4 is a view of the cusp curve of the envelope shown in FIG. 3A;

FIG. 5 is the locus of the possible source points which approach theobservation point P along the radiation direction with the wave speed atthe retarded time, a locus that is henceforth referred to as thebifurcation surface of the observer at P;

FIG. 6 is a view of the cross sections of the bifurcation surface andthe source distribution pattern with a cylinder whose axis coincideswith the rotation axis of the source;

FIGS. 7A and 7B are views of two examples of the apparatus of thepresent invention showing the dielectric substrate, the electrodes and asuperluminally moving polarised region of the dielectric substrate;

FIG. 8 is a diagram showing the wave fronts, and the envelope of thewave fronts, of the electromagnetic emission from a particular volumeelement (source point) S within the rectilinearly moving, acceleratingdistribution pattern of the superluminal source generated by the presentinvention; and

FIGS. 9A-9F show the evolution in observation time of the relativepositions and the envelope of a set of wave fronts emitted during alimited interval of (retarded time; the snapshots 9A-9F include times atwhich the envelope has not yet developed a cusp [9A and 9B], has a cusp[9C-9E], and has already lost its cusp 9F.

Prior to description of the invention, it is appropriate to discuss theprinciples underlying it.

Bolotovskii and Ginzburg (Soviet Phys. Usp. 15, 184, 1972) andBolotovskii and Bykov (Sovet Phys. Usp. 33, 477, 1990) have shown thatthe coordinated motion of aggregates of charged particles can give riseto extended electric charges and currents whose distribution patternspropagate with a phase speed exceeding the speed of light in vacuo andthat, once created, such propagating charged patterns act as sources ofthe electromagnetic fields in precisely the same way as any other movingsources of these fields. That the distribution patterns of these sourcestravel faster than light is not, of course, in any way incompatible withthe requirements of special relativity. The superluminally movingpattern is created by the coordinated motion of aggregates ofsubluminally moving particles.

We have solved Maxwell's equations for the electromagnetic field that isgenerated by an extended source distribution pattern of this type in thecase where the charged pattern rotates about a fixed axis with aconstant angular frequency.

There are solutions of the homogeneous wave equation referred to, interalia, as non-diffracting radiation beams, focus wave modes orelectromagnetic missiles, which describe signals that propagate throughspace with unexpectedly slow rates of decay or spreading. The potentialpractical significance of such signals is clearly enormous. The searchfor physically realizable sources of them, however, has so far remainedunsuccessful. Our calculation pinpoints a concrete example of thesources that are currently looked for in this field by establishing aphysically tenable inhomogeneous solution of Maxwell's equations withthe same characteristics.

Investigation of the present emission process was originally motivatedby the observational data on pulsars. The radiation received from thesecelestial sources of radio waves consists of highly coherent pulses(with as high a brightness temperature as 10³⁰° K) which recurperiodically (with stable periods of the order of 1 sec). The intensemagnetic field (˜10¹² G) of the central neutron star in a pulsar affectsa coupling between the rotation of this star and that of thedistribution pattern of the plasma surrounding it, so that themagnetospheric charges and currents in these objects are of the sametype as those described above. The effect responsible for the extremedegree of coherence of the observed emission from pulsars, therefore,may well be the violation of the inverse square law that is herepredicted by our calculation.

The present analysis is relevant also to the mathematically similarproblem of the generation of acoustic radiation by supersonic propellersand helicopter rotors, although this is not discussed in detail here.

We begin by considering the waves that are emitted by an element of thedistribution pattern of the superluminally rotating source from thestandpoint of geometrical optics. Next, we calculate the amplitudes ofthese waves, i.e. the Green's function for the problem, from theretarded potential. We then specify the bifurcation surface of theobserver and proceed to calculate the electromagnetic radiation arisingfrom an extended source with a superluminally moving distributionpattern. The singularities of the integrands of the radiation integralsthat occur on the bifurcation surface are here handled by means of thetheory of generalised functions: the electric and magnetic fields aregiven by the Hadamard's finite parts of the divergent integrals thatresult from differentiating the retarded potential under the integralsign. The theory is then concluded with a descriptive account of theanalysed emission process in more physical terms, the description ofexamples of the apparatus, and an outline of the applications of theinvention.

I. Envelope of the Wave Fronts and its Cusp

Consider a point source (an element of the propagating distributionpattern of a volume source) which moves on a circle of radius r with theconstant angular velocity ωê_(z), i.e. whose path x(t) is given, interms of the cylindrical polar coordinates (r,φ,z), by

r=const., z=const., φ={circumflex over (φ)}+ωt,   (1)

where ê_(z) is the basis vector associated with z, and {circumflex over(φ)} the initial value of φ.

The wave fronts that are emitted by this point source in an empty andunbounded space are described by

|x _(P) −x(t)|=c(t _(P) −t),   (2)

where the constant c denotes the wave speed, and the coordinates (x_(P),t_(P))=(r_(P),φ_(P),z_(P),t_(P)) mark the spacetime of observationpoints. The distance R between the observation point x_(P) and a sourcepoint x is given by

$\begin{matrix}{{{\left. {x_{P} - x} \right\rbrack \equiv {R(\phi)}} = \left\lbrack {\left( {z_{P} - z} \right)^{2} + r_{P}^{2} + r^{2} - {2r_{P}r\; {\cos \left( {\phi_{P} - \phi} \right)}}} \right\rbrack^{\frac{1}{2}}},} & (3)\end{matrix}$

so that inserting (1) in (2) we obtain

$\begin{matrix}{{{R(t)} \equiv \left\lbrack {\left( {z_{P} - z} \right)^{2} + r_{P}^{2} + r^{2} - {2r_{p}r\; {\cos \left( {\phi_{P} - \hat{\phi} - {\omega \; t}} \right)}}} \right\rbrack^{\frac{1}{2}}} = {{c\left( {t_{P} - t} \right)}.}} & (4)\end{matrix}$

These wave fronts are expanding spheres of radii c(t_(P)−t) whose fixedcentres (r_(P)=r_(P),φ_(P)={circumflex over (φ)}+ωt,z_(P)=z) depend ontheir emission times t (see FIG. 1).

Introducing the natural length scale of the problem, c/ω and usingt_(P)=(φ−{circumflex over (φ)})/ω to eliminate t in favour of φ, we canexpress (4) in terms of dimensionless variables as

g≡φ−φ _(P) +{circumflex over (R)}(φ)=φ,   (5)

in which {circumflex over (R)}≡Rω/c, and

φ≡{circumflex over (φ)}−{circumflex over (φ)}_(P)   (6)

stands for the difference between the positions {circumflex over(φ)}=φ−ωt of the source point and {circumflex over (φ)}_(p)≡φ_(p)−ωt_(p)of the observation point in the (r, {circumflex over (φ)}, z)-space. TheLagrangian coordinate {circumflex over (φ)} in (5) lies within aninterval of length 2π (e.g. −π<{circumflex over (φ)}≦π), while the angleφ, which denotes the azimuthal position of the source point at theretarded time t, ranges over (−∞, ∞).

FIG. 1 depicts the wave fronts described by (5) for fixed values of(r,{circumflex over (φ)},z) and of φ (or t_(p)), and a discrete set ofvalues of φ (or t). [In this figure, the heavier curves show the crosssection of the envelope with the plane of the orbit of the sourcedistribution pattern. The larger of the two dotted circles designatesthe orbit (at r=3c/ω) and the smaller the light cylinder (r_(p)=c/ω).]

These wave fronts possess an envelope because when r>c/ω, and so thespeed of the source distribution pattern exceeds the wave speed, severalwave fronts with differing emission times can pass through a singleobservation point simultaneously. Or stated mathematically, for certainvalues of the coordinates (r_(p), {circumflex over (φ)}_(p), z_(p); r,z) the function g(φ) shown in FIG. 2 is oscillatory and so can equal øat more than one value of the retarded position φ: a horizontal lineφ=constant intersects the curve (a) in FIG. 2 at either one or threepoints. [FIG. 2 is drawn for φ_(p)0, {circumflex over (r)}_(p)=3,{circumflex over (r)}=2 and (a) {circumflex over (z)}={circumflex over(z)}_(p), inside the envelope, (b) {circumflex over (z)}={circumflexover (z)}_(c), on the cusp curve of the envelope, (c) {circumflex over(z)}=2{circumflex over (z)}_(c)−{circumflex over (z)}_(p), outside theenvelope. The marked adjacent turning points of curve (a) have thecoordinates ((φ_(±), φ_(±)), and φ_(out) represents the solution ofg(φ)=φ₀ for a φ₀ that tends to φ⁻ from below.]

Wave fronts become tangent to one another and so form an envelope atthose points (r_(p), {circumflex over (φ)}_(p), z_(p)) for which tworoots of g(φ)=φ coincide. The equation describing this envelope cantherefore be obtained by eliminating φ between g=φ and ∂_(g)/∂φ=0

Thus, the values of φ on the envelope of the wave fronts are given by

∂g/∂φ=1−{circumflex over (r)}{circumflex over (r)} _(P)sin(φ_(P)−φ)/{circumflex over (R)}(φ)=0.   (7)

When the curve representing g(φ) is as in (a) of FIG. 2 (i.e.{circumflex over (r)}>1 and Δ>0), equation this has the doubly infiniteset of solutions φ=φ±+2nπ, where

$\begin{matrix}{{\phi_{\pm} = {\phi_{P} + {2\pi} - {\arccos\left\lbrack {\left( {1 \mp \Delta^{\frac{1}{2}}} \right)/\left( {\hat{r}{\hat{r}}_{P}} \right)} \right\rbrack}}},} & (8) \\{{\Delta \equiv {{\left( {{\hat{r}}_{P}^{2} - 1} \right)\left( {{\hat{r}}^{2} - 1} \right)} - \left( {\hat{z} - {\hat{z}}_{P}} \right)^{2}}},} & (9)\end{matrix}$

n is an integer, and ({circumflex over (r)}, {circumflex over(z)},{circumflex over (r)}_(p),{circumflex over (z)}_(p)) stand for thedimensionless coordinates rω/c, zω/c, r_(p)cω/c and z_(p)ω/c,respectively. The function g(φ) is locally maximum at φ₊2nπ and minimumat φ. +2nπ.

Inserting φ=φ±(5) and solving the resulting equation for ø as a functionof ({circumflex over (r)}_(p), {circumflex over (z)}_(p)), we find thatthe envelope of the wave fronts is composed of two sheets:

$\begin{matrix}{{\varphi = {{\varphi_{\pm} \equiv {g\left( \phi_{\pm} \right)}} = {{2\pi} - {\arccos\left\lbrack {\left( {1 \mp \Delta^{\frac{1}{2}}} \right)/\left( {\hat{r}{\hat{r}}_{P}} \right)} \right\rbrack} + {\hat{R}}_{\pm}}}},} & (10)\end{matrix}$

in which

$\begin{matrix}{{\hat{R}}_{\pm} \equiv \left\lbrack {\left( {\hat{z} - {\hat{z}}_{P}} \right)^{2} + {\hat{r}}^{2} + {\hat{r}}_{P}^{2} - {2\left( {1 \mp \Delta^{\frac{1}{2}}} \right)}} \right\rbrack^{\frac{1}{2}}} & (11)\end{matrix}$

are the values of {circumflex over (R)} at φ=φ_(±). For a fixed sourcepoint (r, {circumflex over (φ)}, z), equation (10) describes a tube-likespiralling surface in the (r_(p), {circumflex over (φ)}_(p),z_(p))-space of observation points that extends from the speed-of-lightcylinder {circumflex over (r)}_(p)=1 to infinity. [A three-dimensionalview of the light cylinder and the envelope of the wave fronts for thesame source point (S) as that in FIG. 1 is presented in FIG. 3A (onlythose parts of these surfaces are shown which lie within the cylindricalvolume {circumflex over (r)}_(p)≦9, −2.25≦{circumflex over(z)}_(p)−{circumflex over (z)}≦2.25).]

The two sheets φ=φ_(±) of this envelope meet at a cusp. The cusp occursalong the curve

$\begin{matrix}{{\varphi = {{{2\pi} - {\arccos \left\lbrack {1/\left( {\hat{r}{\hat{r}}_{P}} \right)} \right\rbrack} + \left( {{{\hat{r}}_{P}^{2}{\hat{r}}^{2}} - 1} \right)^{\frac{1}{2}}} \equiv \varphi_{c}}},} & \left( {12a} \right) \\{{\hat{z} = {{{\hat{z}}_{P} \pm {\left( {{\hat{r}}_{P}^{2} - 1} \right)^{\frac{1}{2}}\left( {{\hat{r}}^{2} - 1} \right)^{\frac{1}{2}}}} \equiv {\hat{z}}_{c}}},} & \left( {12b} \right)\end{matrix}$

shown in FIG. 4 and constitutues the locus of points at which threedifferent wave fronts intersect tangentially. [FIG. 4 depicts thesegment −15≦{circumflex over (z)}_(P)−{circumflex over (z)}≦15 of thecusp curve of the envelope shown in FIG. 3A. This curve touches—and istangent to—the light cylinder at the point ({circumflex over (r)}_(P)=1,{circumflex over (z)}_(P)={circumflex over (z)},φ=φ_(c)|_({circumflex over (r)}) _(P) ₌₁) on the plane of the orbit.]

On the cusp curve φ=φ_(c), z=z_(c), the function g(φ) has a point ofinflection [(b) of FIGS. 2] and ∂² _(g)/∂φ², as well as ∂_(g)/∂φ, and gvanish at

φ=φ_(P)+2π−arccos [1/({circumflex over (r)}{circumflex over (r)}_(P))]≡φ_(c),   (12c)

This, in conjunction with t=(φ−{circumflex over (φ)})/ω, represents thecommon emission time of the three wave fronts that are mutuallytangential at the cusp curve of the envelope.

In the highly superluminal regime, where {circumflex over (r)}>>1, theseparation of the ordinates φ₊ and φ⁻ of adjacent maxima and minima in(a) of FIG. 2 can be greater than 2π. A horizontal line φ=constant willthen intersect the curve representing g(φ) at more than three points,and so give rise to simultaneously received contributions that are madeat 5, 7, . . . , distinct values of the retarded time. In such cases,the sheet φ⁻ of the envelope (issuing from the conical apex of thissurface) undergoes a number of intersections with the sheet φ₊ beforereaching the cusp curve. We shall be concerned in this paper, however,mainly with source elements whose distances from the rotation axis donot appreciably exceed the radius c/w of the speed-of-light cylinder andso for which the equation g(φ)=φ has at most three solutions.

At points of tangency of their fronts, the waves which interfereconstructively to form the envelope propagate normal to the sheetsφ=φ_(±) (r_(p), z_(p)) of this surface, in the directions

$\begin{matrix}\begin{matrix}{{\hat{n}}_{\pm} \equiv {\left( {c/\omega} \right){\nabla_{P}\left( {\varphi_{\pm} - \varphi} \right)}}} \\{{= {{{{\hat{e}}_{rp}\left\lbrack {{\hat{r}}_{P} - {{\hat{r}}_{P}^{- 1}\left( {1 \mp \Delta^{\frac{1}{2}}} \right)}} \right\rbrack}/{\hat{R}}_{\pm}} + {{\hat{e}}_{\phi \; p}/{\hat{r}}_{P}} + {{{\hat{e}}_{zp}\left( {{\hat{z}}_{P} - \hat{z}} \right)}/{\hat{R}}_{\pm}}}},}\end{matrix} & (13)\end{matrix}$

with the speed c. (ê_(r p), êφ_(p) and ê_(z p) are the unit vectorsassociated with the cylindrical coordinates r_(p), φ_(p) and z_(p) ofthe observation point, respectively.) Nevertheless, the resultingenvelope is a rigidly rotating surface whose shape does not change withtime: in the (r_(p), {circumflex over (φ)}_(p) and z_(p))-space, itsconical apex is stationary at (r, {circumflex over (φ)}, z), and itsform and dimensions only depend on the constant parameter {circumflexover (r)}.

The set of waves that superpose coherently to form a particular sectionof the envelope or its cusp, therefore, cannot be the same (i.e. cannothave the same emission times) at different observation times. The packetof focused waves constituting any given segment of the cusp curve of theenvelope, for instance, is constantly dispersed and reconstructed out ofother waves. This one-dimensional caustic would not be unlimited in itsextent, as shown in FIG. 4, unless the source distribution pattern isinfinitely long-lived: only then would the duration of the sourcedistribution pattern encompass the required intervals of emission timefor every one of its constituent segments.

II. Amplitudes of the Waves Generated by a Point Source

Our discussion has been restricted so far to the geometrical features ofthe emitted wave fronts. In this section we proceed to find theLienard-Wechert potential for these waves.

The scalar potential arising from a volume element of the movingdistribution pattern of the source we have been considering is given bythe retarded solution of the wave equation

∇′² G ₀−∂² G ₀/∂(ct′)²=−4πρ₀,   (14a)

in which

ρ₀(r′, φ′, z′, t′)=δ(r′−r)δ(φ′−ωt′−{circumflex over(φ)})δ(z′−z)/r′  (14b)

is the density of a point source of unit strength with the trajectory(1). In the absence of boundaries, therefore, this potential has thevalue

$\begin{matrix}{{{G_{0}\left( {x_{P},t_{P}} \right)} = {\int{d^{3}x^{\prime}{dt}^{\prime}{\rho_{0}\left( {x^{\prime},t^{\prime}} \right)}{{\delta \left( {t_{P} - t^{\prime} - {{{x_{P} - x^{\prime}}}/c}} \right)}/{{x_{P} - x^{\prime}}}}}}}\mspace{25mu}} & {\left( {15a} \right)} \\{{= {\int_{- \infty}^{+ \infty}{{dt}^{\prime}{{\delta \left\lbrack {t_{P} - {t^{\prime}{{R\left( t^{\prime} \right)}/c}}} \right\rbrack}/{R\left( t^{\prime} \right)}}}}},} & {\left( {15b} \right)}\end{matrix}$

where R(t′) is the function defined in (4) (see e.g. Jackson, ClassicalElectrodynamics, Wiley, New York 1975).

If we use (1) to change the integration variable t′ in (15b) to φ, andexpress the resulting integrand in terms of the quantities introduced in(3), (5) and (6), we arrive at

G ₀(r, r _(P), {circumflex over (φ)}−{circumflex over (φ)}_(P) , z−z_(P))=∫_(−∞) ^(+∞) dφδ[g(φ)−φ]/R(φ).   (16)

This can then be rewritten, by formally evaluating the integral, as

$\begin{matrix}{{G_{0} = {\sum\limits_{\phi = \phi_{i}}^{\;}\frac{1}{R{{{\partial g}/{\partial\phi}}}}}},} & (17)\end{matrix}$

where the angles φ_(j) are the solutions of the transcendental equationg(φ)=φ in −∞<φ<+∞ and correspond, in conjunction with (1), to theretarded times at which the source point (r, {circumflex over (φ)}, z)makes its contribution towards the value of G₀ at the observation point(r_(p), {circumflex over (φ)}_(p), z_(p)).

Equation (17) shows, in the light of FIG. 2, that the potential G₀ of apoint source is discontinuous on the envelope of the wave fronts: if weapproach the envelope from outside, the sum in (17) has only a singleterm and yields a finite value for G₀, but if we approach this surfacefrom inside, two of the φ_(j) s coalesce at an extremum of g and (17)yields a divergent value for G₀. Approaching the sheet φ=φ₊ or φ=φ⁻ ofthe envelope from inside this surface corresponds, in FIG. 2, to raisingor lowering a horizontal line φ=φ₀=const., with φ⁻≦φ₀≦φ₊ until itintersects the curve (a) of this figure at its maximum or minimumtangentially. At an observation point thus approached, the sum in (17)has three terms, two of which tend to infinity.

On the other hand, approaching a neighbouring observation point justoutside the sheet φ=φ⁻ (say) of the envelope corresponds, in FIG. 2, toraising a horizontal line φ=φ₀=const., with φ₀≦φ⁻ towards a limitingposition in which it tends to touch curve (a) at its minimum. So long asit has not yet reached the limit, such a line intersects curve (a) atone point only. The equation g(φ)=φ therefore has only a single solutionφ=φ_(out) in this case which is different from both φ₊ and φ⁻ and so atwhich ∂_(g)/∂φ is non-zero (see FIG. 2). The contribution that thesource distribution pattern makes when located at φ=φ_(out) is receivedby both observers, but the constructively interfering waves that areemitted at the two retarded positions approaching φ. only reach theobserver inside the envelope.

The function G₀ has an even stronger singularity at the cusp curve ofthe envelope. On this curve, all three of the φ_(j)s coalesce [(b) ofFIG. 2] and each denominator in the expression in (17) both vanishes andhas a vanishing derivative (∂_(g)/∂φ=∂² _(g)/∂φ²=0).

There is a standard asymptotic technique for evaluating radiationintegrals with coalescing critical points that describe caustics. Byapplying this technique—which we have outlined in Appendix A—to theintegral in (16), we can obtain a uniform asymptotic approximation to G₀for small |φ₊−φ⁻|, i.e. for points close to the cusp curve of theenvelope where G₀ is most singular. The result is

$\begin{matrix}{{{\left. G_{0}^{i\; n} \right.\sim 2}{{c_{1}^{- 2}\left( {1 - ^{2}} \right)}^{- \frac{1}{2}}\left\lbrack {{p_{0}{\cos \left( {\frac{1}{3}\arcsin \; } \right)}} - {c_{1}q_{0}{\sin \left( {\frac{2}{3}\arcsin \; } \right)}}} \right\rbrack}},\mspace{76mu} {{} < 1},} & (18) \\{\mspace{76mu} {and}} & \; \\{{\left. G_{0}^{out} \right.\sim{c_{1}^{- 2}\left( {^{2} - 1} \right)}^{- \frac{1}{2}}}{\quad{\left\lbrack {{p_{0}{\sinh \left( {\frac{1}{3}{arc}\; \cosh {}} \right)}} + {c_{1}q_{0}{{sgn}()}{\sinh \left( {\frac{2}{3}{arc}\; \cosh {}} \right)}}} \right\rbrack,\mspace{79mu} {{} > 1},}}} & (19)\end{matrix}$

where c₁, p₀, q₀ and X are the functions of (r, z) defined in (A2),(A5), (A6) and (A10), and approximated in (A23)-(A30). The superscripts‘in’ and ‘out’ designate the values of G₀ inside and outside theenvelope, and the variable X equals +1 and −1 on the sheets φ=φ₊ andφ=φ⁻ of this surface, respectively.

The function G₀ ^(out) is indeterminate but finite on the envelope [cf.(A39)], whereas G₀ ^(in) diverges like

${\sqrt{3}{{c_{1}^{- 2}\left( {p_{0} \mp {c_{1}q_{0}}} \right)}/\left( {1 - ^{2}} \right)^{\frac{1}{2}}}\mspace{14mu} {as}\mspace{14mu} }->{\pm 1.}$

The singularity structure of G₀ ^(in) close to the cusp curve isexplicitly exhibited by

$\begin{matrix}{{{\left. G_{0}^{i\; n} \right.\sim\frac{2}{3^{\frac{1}{6}}}}\left( {\omega/c} \right)\left( {{{\hat{r}}^{2}{\hat{r}}_{P}^{2}} - 1} \right)^{- \frac{1}{2}}{{c_{0}^{\frac{1}{2}}\left( {{\hat{z}}_{c} - \hat{z}} \right)}^{\frac{1}{2}}/\left\lbrack {{c_{0}^{3}\left( {{\hat{z}}_{c} - \hat{z}} \right)}^{3} - \left( {\varphi_{c} - \varphi} \right)^{2}} \right\rbrack^{\frac{1}{2}}}},} & (20)\end{matrix}$

in which 0≦{circumflex over (z)}_(e)−{circumflex over (z)}<<1,|φ_(e)−φ|<<1 and

$\begin{matrix}{c_{0} \equiv {\frac{2}{3^{\frac{2}{3}}}\left( {{{\hat{r}}^{2}{\hat{r}}_{P}^{2}} - 1} \right)^{- 1}\left( {{\hat{r}}_{P}^{2} - 1} \right)^{\frac{1}{2}}\left( {{\hat{r}}^{2} - 1} \right)^{\frac{1}{2}}}} & (21)\end{matrix}$

[see (18) and (A22)-(A26)]. It can be seen from this expression thatboth the singularity on the envelope (at which the quantity inside thesquare brackets vanishes) and the singularity at the cusp curve (atwhich {circumflex over (z)}_(c)−{circumflex over (z)} and φ_(c)−φvanish) are integrable singularities.

The potential of a volume source, which is given by the superposition ofthe potentials G₀ of its constituent volume elements, and so involvesintegrations with respect to (r, {circumflex over (φ)}, z), is thereforefinite. Since they are created by the coordinated motion of aggregatesof particles, the types of source distribution patterns we have beenconsidering cannot, of course, be point-like. It is only in thephysically unrealizable case where the distribution pattern of asuperluminal source is point-like that its potential has the extendedsingularities described above.

In fact, not only is the potential of an extended source with asuperluminally moving distribution pattern singularity free, but itdecays in the far zone like the potential of any other source. Thefollowing alternative form of the retarded solution to the wave equation∇²A₀−∂²A₀/∂(ct)²=−4πρ[which may be obtained from (15a) by performing theintegration with respect to time]:

A ₀ =∫d ³ xρ(x, t _(P) −|x−x _(P) |/c)/|x−x _(P)|  (22)

shows that if the density ρ of the source is finite and vanishes outsidea finite volume, then the potential A₀ decays like |x_(p)|⁻¹ as thedistance |x_(p)−x|≅|x_(p)| of the observer from the source tends toinfinity.

III. The Bifurcation Surface of an Observer

Let us now consider an extended source distribution pattern whichrotates about the z-axis with the constant angular frequency ω. Thedensity of such a source—when it has a distribution with an unchangingpattern—is given by

ρ(r,φ,z,t)=ρ(r,{circumflex over (φ)},z),   (23)

where the Lagrangian variable {circumflex over (φ)} is defined by φ−ωtas in (1), and ρ can be any function of (r, {circumflex over (φ)}, z)that vanishes outside a finite volume.

If we insert this density in the expression for the retarded scalarpotential and change the variables of integration from (r,{circumflexover (φ)},z,t) to (r,{circumflex over (φ)},z,t), we obtain

$\begin{matrix}{{{A_{0}\left( {x_{P},t_{P}} \right)} = {\int{d^{3}{xdt}\; {\rho \left( {x,t} \right)}{{\delta \left( {t_{P} - t - {{{x - x_{P}}}/c}} \right)}/{{x - x_{P}}}}}}}\mspace{34mu}} & {\left( {24a} \right)} \\{{= {\int{{rdrd}\; \hat{\phi}\; {dz}\; {\rho \left( {r,\hat{\phi},z} \right)}{G_{0}\left( {r,r_{P},{\hat{\phi} - {\hat{\phi}}_{P}},{z - z_{P}}} \right)}}}},} & {\left( {24b} \right)}\end{matrix}$

where G₀ is the function defined in (16) which represents the scalarpotential of a corresponding point source. That the potential of theextended source distribution pattern in question is given by thesuperposition of the potentials of the moving source points thatconstitute the distribution pattern is an advantage that is gained bymarking the space of source points with the natural coordinates(r,{circumflex over (φ)},z) of the source distribution pattern. Thisadvantage is lost if we use any other coordinates.

In Sec. II, where the distribution pattern of the source was point-like,the coordinates (r,{circumflex over (φ)},z) of the source point in G₀(r,r_(p), {circumflex over (φ)}−{circumflex over (φ)}_(p), z−z_(p)) wereheld fixed and we were concerned with the behaviour of this potential asa function of the coordinates (r_(p), {circumflex over (φ)}_(p), z_(p))of the observation point. When we superpose the potentials of the volumeelements that constitute an extended source distribution pattern, on theother hand, the coordinates (r_(p), {circumflex over (φ)}_(p), z_(p))are held fixed and we are primarily concerned with the behaviour of G₀as a function of the integration variables (r, {circumflex over (φ)},z).

Because G₀ is invariant under the interchange of (r, {circumflex over(φ)}, z) and (r_(P), {circumflex over (φ)}_(P), z_(P)) if φ is at thesame time changed to −φ[see (5) and (16)], the singularity of G₀ occurson a surface in the (r, {circumflex over (φ)}, z)-space of source pointswhich has the same shape as the envelope shown in FIG. 3A but issuesfrom the fixed point (r_(P), {circumflex over (φ)}_(P), z_(P)) andspirals around the z-axis in the opposite direction to the envelope.[FIG. 5 in which the light cylinder and the bifurcation surfaceassociated with the observation point P are shown for a counterclockwisemotion of the source distribution pattern. In this figure P is locatedat {circumflex over (r)}_(P)=9, and only those parts of these surfacesare shown which lie within the cylindrical volume {circumflex over(r)}≦11, −1.5≦{circumflex over (z)}−{circumflex over (z)}_(P)≦1.5. Thetwo sheets φ=φ_(±)(r,z) of the bifurcation surface meet along a cusp (acurve of the same shape as that shown in FIG. 4) that is tangent to thelight cylinder. For an observation point in the far zone ({circumflexover (r)}_(P)>>1), the spiralling surface that issues from P undergoes alarge number of turns—in which its two sheets intersect oneanother—before reaching the light cylinder.]

In this paper, we refer to this locus of singularities of G₀ as thebifurcation surface of the observation point P.

Consider an observation point P for which the bifurcation surfaceintersects the source distribution pattern, as in FIG. 6. [In FIG. 6,the full curves depict the cross section, with the cylinder {circumflexover (r)}=1.5, of the bifurcation surface of an observer located at{circumflex over (r)}_(p)=3. (The motion of the source distributionpattern is counterclockwise.) Projection of the cusp curve of thisbifurcation surface onto the cylinder {circumflex over (r)}=1.5 is shownas a dotted curve, and the region occupied by the source distributionpattern as a dotted area. In this figure the observer's position is suchthat one of the points (φ=φ_(c), z=z_(c)) at which the cusp curve inquestion intersects the cylinder {circumflex over (r)}=1.5—the one withz_(c)>0—is located within the source distribution pattern. As the radialposition r_(p) of the observation point tends to infinity, theseparation—at a finite distance z_(c)−z from ((φ_(c), z_(c))—of theshown cross sections decreases like

$\left. {r_{p}^{- \frac{3}{2}}.} \right\rbrack$

The envelope of the wave fronts emanating from a volume element of thepart of the source distribution pattern that lies within thisbifurcation surface encloses the point P, but P is exterior to theenvelope associated with an element of the source distribution patternthat lies outside the bifurcation surface.

We have seen that three wave fronts—propagating in differentdirections—simultaneously pass an observer who is located inside theenvelope of the waves emanating from a point source, and only onewavefront passes an observer outside this surface. Hence, in contrast tothe elements of the source distribution pattern outside the bifurcationsurface which influence the potential at P at only a single value of theretarded time, this potential receives contributions from each of theelements inside the bifurcation surface at three distinct values of theretarded time.

The elements of the source distribution pattern inside but adjacent tothe bifurcation surface, for which G₀ diverges, are sources of theconstructively interfering waves that not only arrive at Psimultaneously but also are emitted at the same (retarded) time. Theseelements of the source distribution pattern approach the observer alongthe radiation direction x_(p)−x with the wave speed at the retardedtime, i.e. are located at distances R(t) from the observer for which

$\begin{matrix}{{\frac{dR}{dt}}_{t = {t_{P} - {R/c}}} = {- c}} & (25)\end{matrix}$

[see (4), (7) and (8)]. Their accelerations at the retarded time,

$\begin{matrix}{{{\frac{d^{2}R}{{dt}^{2}}}_{t = {t_{P} - {R/c}}} = {\mp \frac{c\; {\omega\Delta}^{\frac{1}{2}}}{{\hat{R}}_{\pm}}}},} & (26)\end{matrix}$

are positive on the sheet φ=φ⁻ of the bifurcation surface and negativeon φ=φ₊.

The elements of the source distribution pattern on the cusp curve of thebifurcation surface, for which Δ=0 and all three of the contributingretarded times coincide, approach the observer—according to (26)—withzero acceleration as well as with the wave speed.

From a radiative point of view, the most effective volume elements ofthe distribution pattern of the superluminal source in question arethose that approach the observer along the radiation direction with thewave speed and zero acceleration at the retarded time, since the ratioof the emission to reception time intervals for the waves that aregenerated by these particular elements of the source distributionpattern generally exceeds unity by several orders of magnitude. On eachconstituent ring of the source distribution pattern that lies outsidethe light cylinder (r=c/ω) in a plane of rotation containing theobservation point, there are two volume elements that approach theobserver with the wave speed at the retarded time: one whose distancefrom the observer diminishes with positive acceleration, and another forwhich this acceleration is negative. These two elements are closer toone another the smaller the radius of the ring. For the smallest of suchconstituent rings, i.e. for the one that lies on the light cylinder, thetwo volume elements in question coincide and approach the observer alsowith zero acceleration.

The other constituent rings of the source distribution pattern (those onthe planes of rotation which do not pass through the observation point)likewise contain two such elements if their radii are large enough fortheir velocity rωe_(φ) to have a component along the radiation directionequal to c. On the smallest possible ring in each plane, there is againa single volume element—at the limiting position of the two coalescingvolume elements of the neighbouring larger rings—that moves towards theobserver not only with the wave speed but also with zero acceleration.

For any given observation point P, the efficiently radiating pairs ofvolume elements on various constituent rings of the source distributionpattern collectively form a surface: the part of the bifurcation surfaceassociated with P which intersects the source distribution pattern. Thelocus of the coincident pairs of volume elements, which is tangent tothe light cylinder at the point where it crosses the plane of rotationcontaining the observer, constitutes the segment of the cusp curve ofthis bifurcation surface that lies within the source distributionpattern.

Thus the bifurcation surface associated with any given observation pointdivides the volume of the source distribution pattern into two sets ofelements with differing influences on the observed field. As in (18) and(19), the potentials G₀ ^(in) and G₀ ^(out) of the source distributionpattern's elements inside and outside the bifurcation surface havedifferent forms: the boundary |χ(r, r_(P), {circumflex over(φ)}−{circumflex over (φ)}_(P), z−z_(P))|=1 between the domains ofvalidity of (18) and (19) delineates the envelope of wave fronts whenthe source point (r, {circumflex over (φ)}, z) is fixed and thecoordinates (r_(p), {circumflex over (φ)}_(p), z_(p)) of the observationpoint are variable, and describes the bifurcation surface when theobservation point (r_(p), {circumflex over (φ)}_(p), z_(p)) is fixed andthe coordinates (r, {circumflex over (φ)}, z) of the source point sweepa volume.

The expression (24b) for the scalar potential correspondingly splitsinto the following two terms when the observation point is such that thebifurcation surface intersects the source distribution pattern:

$\begin{matrix}{A_{0} = {\int{{dV}_{\rho}G_{0}}}} & {{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\left( {27a} \right)} \\{{= {{\int_{V_{in}}^{\;}{{dV}_{\rho}G_{0}^{in}}} + {\int_{V_{out}}^{\;}{{dV}_{\rho}G_{0}^{out}}}}}\ ,} & {\left( {27b} \right)}\end{matrix}$

where dV ≡ rdrd{circumflex over (φ)} dz, V_(in) and V_(out) designatethe portions of the source distribution pattern which fall inside andoutside the bifurcation surface (see FIG. 6), and G₀ ^(in) and G₀ ^(out)denote the different expressions for G₀ in these two regions.

Note that the boundaries of the volume V_(in) depend on the position(r_(p), {circumflex over (φ)}_(p), z_(p)) of the observer: the parameter{circumflex over (r)}_(p) fixes the shape and size of the bifurcationsurface, and the position (r_(p), {circumflex over (φ)}_(p), z_(p)) ofthe observer specifies the location of the conical apex of this surface.When the observation point is such that the cusp curve of thebifurcation surface intersects the source distribution pattern, thevolume V_(in) is bounded by φ=φ., φ=φ₊ and the part of the boundary ρ(r,{circumflex over (φ)}, z)=0 of the distribution pattern that fallswithin the bifurcation surface. The corresponding volume V_(out) isbounded by the same patches of the two sheets of the bifurcation surfaceand by the remainder of the boundary of the source distribution pattern.

In the vicinity of the cusp curve (12), i.e. for |φ_(c)−φ|<<1 and0≦{circumflex over (z)}_(c)−{circumflex over (z)}<<1, the cross sectionof the bifurcation surface with a cylinder {circumflex over(r)}=constant is described by

$\begin{matrix}{{\varphi_{\pm} - \varphi_{c}} \simeq {{{- \left( {{\hat{r}}^{2} - 1} \right)^{\frac{1}{2}}}\left( {{\hat{r}}_{P}^{2} - 1} \right)^{\frac{1}{2}}\left( {{{\hat{r}}^{2}{\hat{r}}_{P}^{2}} - 1} \right)^{- \frac{1}{2}}\left( {{\hat{z}}_{c} - \hat{z}} \right)} \pm {\frac{2^{\frac{3}{2}}}{3}\left( {{\hat{r}}^{2} - 1} \right)^{\frac{3}{4}}\left( {{\hat{r}}_{P}^{2} - 1} \right)^{\frac{3}{4}}\left( {{{\hat{r}}_{P}^{2}{\hat{r}}^{2}} - 1} \right)^{- \frac{3}{2}}\left( {{\hat{z}}_{c} - \hat{z}} \right)^{\frac{3}{2}}}}} & (28)\end{matrix}$

[see (10)-(12) and (A26)]. This cross section, which is shown in FIG. 6,has two branches meeting at the intersections of the cusp curve with thecylinder {circumflex over (r)}=constant whose separation φ—at a given({circumflex over (z)}_(c)−{circumflex over (z)})—diminishes like

${\hat{r}}_{P}^{- \frac{3}{2}}$

in the limit {circumflex over (r)}_(p)→∞. Thus, at finite distances{circumflex over (z)}_(c)−{circumflex over (z)} from the cusp curve, thetwo sheets φ=φ⁻ and φ=φ₊ of the bifurcation surface coalesce and becomecoincident with the surface

$\varphi = \left. {{\frac{1}{2}\left( {\varphi_{-} + \varphi_{+}} \right)} \equiv {c_{2}\mspace{14mu} {as}\mspace{14mu} {\hat{r}}_{P}}}\rightarrow{\infty.} \right.$

That is to say, the volume V_(in) vanishes like

${\hat{r}}_{P}^{- \frac{3}{2}}.$

Because the dominant contributions towards the value of the radiationfield come from those source distribution pattern's elements thatapproach the observer—along the radiation direction—with the wave speedand zero acceleration at the retarded time, in what follows, we shall beprimarily interested in far-field observers the cusp curves of whosebifurcation surfaces intersect the source distribution pattern. For suchobservers, the Green's function

$\lim\limits_{{\hat{r}}_{p}\rightarrow\infty}G_{0}$

undergoes a jump discontinuity across the coalescing sheets of thebifurcation surface: the values of X on the sheets φ=φ_(±) and hence thefunctions G₀ ^(out)|φ=φ⁻ and G₀ ^(out)|φ=φ₊ remain different even in thelimit where φ=φ⁻ and φ=φ⁻ coincide [cf. (A10) and (A39)].

IV. Derivatives of the Radiation Integrals and Their Hadamard's FiniteParts A. Gradient of the Scalar Potential

In this section we begin the calculation of the electric and magneticfields by finding the gradient of the scalar potential A₀, i.e. bycalculating the derivatives of the integral in (27a) with respect to thecoordinates (r_(p), {circumflex over (φ)}_(p), z_(p)) of the observationpoint.

If we regard its singular kernel G₀ as a classical function, then theintegral in (27a) is improper and cannot be differentiated under theintegral sign without characterizing and duly handling the singularitiesof its integrand. On the other hand, if we regard G₀ as a generalizedfunction, then it would be mathematically permissible to interchange theorders of differentiation and integration when calculating .gradient∇_(P)A₀.

This interchange results in a new kernel .gradient ∇_(P)G₀ whosesingularities are non-integrable. However, the theory of generalizedfunctions prescribes a well-defined procedure for obtaining thephysically relevant value of the resulting divergent integral, aprocedure involving integration by parts which extracts the so-calledHadamard's finite part of this integral (see e.g. Hoskins, GeneralisedFunctions, Ellis Horwood, London 1979).

Hadamard's finite part of the divergent integral representing∇_(P)A₀.yields the value that we would have obtained if we had firstevaluated the original integral for A₀ as an explicit function of(r_(p), {circumflex over (φ)}_(p), z_(p)) and then differentiated it.From the standpoint of the theory of generalized functions, therefore,differentiation of (27a) yields

∇_(P) A ₀ =∫dV _(ρ)∇_(P) G ₀=(∇_(P) A ₀)_(in)+(∇_(P) A ₀)_(out),   (29a)

in which

(∇_(P)A₀)_(in,out)≡∫_(V) _(in,out) dv_(ρ)∇_(P)G₀ ^(in,out),   (29b)

Since ρ vanishes outside a finite volume, the integral in (27a) extendsover all values of (r,{circumflex over (φ)}, z) and so there is nocontribution from the limits of integration towards the derivative ofthis integral.

The kernels ∇_(P)G₀ ^(in,out) of the above integrals may be obtainedfrom (16). Applying ∇_(P) to the right-hand side of (16) andinterchanging the orders of differentiation and integration, we obtainan integral representation of ∇_(P)G₀ consisting of two terms: onearising from the differentiation of R which decays like r_(p) ⁻² asr_(p)→∞ and so makes no contribution to the field in the radiation zone,and another that arises from the differentiation of the Dirac deltafunction and decays less rapidly than r_(p) ⁻². For an observation pointin the radiation zone, we may discard terms of the order of r_(p) ⁻² andwrite

∇_(P) G ₀≅(ω/c)∫_(−∞) ^(+∞) dφR ⁻¹δ′(g−φ){circumflex over (n)},{circumflex over (r)} _(P)>>1,   (30)

in which δ′ is the derivative of the Dirac delta function with respectto its argument and

{circumflex over (n)}≡ê_(rP)[{circumflex over (r)}_(P)−{circumflex over(r)} cos(φ−φ_(P))]/{circumflex over (R)}+ê_(φP)/{circumflex over(r)}_(P)+ê_(zP)({circumflex over (z)}_(P)−{circumflex over(z)})/{circumflex over (R)}.   (31)

Equation (30) yields ∇_(P)G₀ ^(in) or .gradient ∇_(P)G₀ ^(out) dependingon whether φ lies within the interval (φ⁻, φ₊) or outside it.

If we now insert (30) in (29b) and perform the integrations with respectto {circumflex over (φ)} by parts, we find that

$\begin{matrix}{{\left( {\nabla_{p}A_{0}} \right)_{in} \simeq {\left( {\omega/c} \right){\int_{S}^{\;}{{rdrdz}\left\{ {{- \left\lbrack {\rho G}_{1}^{in} \right\rbrack_{\varphi = \varphi_{-}}^{\varphi = \varphi_{+}}} + {\int_{\varphi_{-}}^{\varphi_{+}}{d\; \varphi {{\partial\rho}/{\partial\hat{\phi}}}\; G_{1}^{in}}}} \right\}}}}},\mspace{79mu} {{\hat{r}}_{P}1},} & (32) \\{\mspace{79mu} {and}} & \; \\{{\left( {\nabla_{p}A_{0}} \right)_{out} \simeq {\left( {\omega/c} \right){\int_{S}^{\;}{{rdrdz}\left\{ {\left\lbrack {\rho \; G_{1}^{out}} \right\rbrack_{\varphi = \varphi_{-}}^{\varphi = \varphi_{+}} + {\left( {\int_{- \pi}^{\varphi_{-}}{+ \int_{\varphi_{+}}^{+ \pi}}} \right)d\; \varphi {{\partial\rho}/{\partial\hat{\phi}}}\; G_{1}^{out}}} \right\}}}}},\mspace{85mu} {{\hat{r}}_{P}1},} & (33)\end{matrix}$

in which S stands for the projection of V_(in) onto the (r, z)-plane,and G₁ ^(in) and G₁ ^(out) are given by the values of

$\begin{matrix}{G_{1} = {{\int_{- \infty}^{+ \infty}{d\; \phi \; R^{- 1}{\delta \left( {g - \varphi} \right)}\hat{n}}} = {\sum\limits_{\phi = \phi_{i}}^{\;}{R^{- 1}{{{\partial g}/{\partial\phi}}}^{- 1}\hat{n}}}}} & (34)\end{matrix}$

for φ inside and outside the interval (φ⁻, φ₊), respectively.

Like G₀ ^(in), the Green's function G₁ ^(in) diverges on the bifurcationsurface φ=φ_(±) where ∂g/∂φ vanishes, but this singularity of G₀ ^(in)is integrable so that the value of the second integral in (32) is finite(see Sec. II and Appendix A). Hadamard's finite part of (∇_(P)A₀)_(in)(denoted by the prefix Fp) is obtained by simply discarding those‘integrated’ or boundary terms in (32) which diverge. Hence, thephysically relevant quantity Fp{(∇_(P)A₀)_(in)} consists—in the farzone—of the volume integral in (32).

Let us choose an observation point for which the cusp curve of thebifurcation surface intersects the source distribution pattern (see FIG.6). When the dimensions (˜L) of the source are negligibly smaller thanthose of the bifurcation surface (i.e. when L<<r_(p) and soz_(c)−z<<r_(p) throughout the source distribution pattern) the functionsG₁ ^(in,out) in (32) and (33) can be approximated by their asymptoticvalues (A34) and (A35) in the vicinity of the cusp curve (see AppendixA).

According to (A34), (A36) and (A44), G₁ ^(in) decays like p₁/c₁ ²=O(1)at points interior to the bifurcation surface where

$\lim\limits_{R_{P}\rightarrow\infty}\chi$

remains finite. since the separation of the two sheets of thebifurcation surface diminishes like

${\hat{r}}_{P}^{- \frac{3}{2}}$

within the source distribution pattern [see (28)], it therefore followsthat the volume integral in (32) is of the order of

${1 \times {\hat{r}}_{P}^{- \frac{3}{2}}},$

a result which can also be inferred from the far-field version of (A34)by explicit integration. Hence,

$\begin{matrix}{{{{Fp}\left\{ \left( {\nabla_{P}A_{0}} \right)_{in} \right\}} = {O\left( {\hat{r}}_{P}^{- \frac{3}{2}} \right)}},{{\hat{r}}_{P}1},} & (35)\end{matrix}$

decays too rapidly to make any contribution towards the value of theelectric field in the radiation zone.

Because G₁ ^(out) is, in contrast to G₁ ^(in), finite on the bifurcationsurface, both the surface and the volume integrals on the right-handside of (33) have finite values. Each component of the second term hasthe same structure as the expression for the potential itself and sodecays like r_(p) ⁻¹ (see the ultimate paragraph of Sec. II). But thefirst term—which would have cancelled the corresponding boundary term in(32) and so would not have survived in the expression for ∇_(P)A₀ hadthe Green's function G₁ been continuous—behaves differently from anyconventional contribution to a radiation field.

Insertion of (A39) in (33) yields the following expression for theasymptotic value of this boundary term in the limit where the observeris located in the far zone and the source distribution pattern islocalized about the cusp curve of his (her) bifurcation surface:

$\begin{matrix}{\left. {{{\int{{{rdrdz}\left\lbrack {\rho \; G_{1}^{out}} \right\rbrack}_{\varphi_{-}}^{\varphi_{+}}\text{∼}\frac{1}{3}{c_{1}}^{- 2}{\int{{rdrdz}\left\lbrack {{p_{1}\left( {\rho {_{\varphi_{+}}{- \rho}}_{\varphi_{-}}} \right)} + {\quad\quad}}\quad \right.}}}}\quad}2c_{1}{q_{1}\left( {\rho {_{\varphi_{+}}{- \rho}}_{\varphi_{-}}} \right)}} \right\rbrack.} & (36)\end{matrix}$

In this limit, the two sheets of the bifurcation surface are essentiallycoincident throughout the domain of integration in (36) [see (28)]. Sothe difference between the values of the source density on these twosheets of the bifurcation surface is negligibly small

$\left( {\text{∼}{\hat{r}}_{P}^{- \frac{3}{2}}} \right)$

for a smoothly distributed source pattern and the functions ρ|φ₃₅appearing in the integrand of (36) may correspondingly be approximatedby their common limiting value ρbs(r, z) on these coalescing sheets.

Once the functions ρ|φ_(±) are approximated by ρbs(r, z) and q₁ by(A41), equation (36) yields an expression which can be written, towithin the leading order in the far-field approximation {circumflex over(r)}_(p)<<1 [see (A44) and (A45)], as

$\begin{matrix}{{{{{\int_{S}^{\;}{{{rdrdz}\left\lbrack {\rho \; G_{1}^{out}} \right\rbrack}_{\varphi_{-}}^{\varphi_{+}}\text{∼}2^{\frac{3}{2}}\left( {c/\omega} \right)^{2}{\hat{r}}_{P}^{- \frac{3}{2}}{\int_{{\hat{r}}_{<}}^{{\hat{r}}_{>}}{d{\hat{r}\left( {{\hat{r}}^{2} - 1} \right)}^{- \frac{1}{4}}n_{1} \times {\quad{\quad{\quad\quad}\quad}\quad}}}}}\quad}{\int_{{\hat{z}}_{c} - {L_{\hat{z}}{\omega/c}}}^{{\hat{z}}_{c}}{d{\hat{z}\left( {{\hat{z}}_{c} - \hat{z}} \right)}^{- \frac{1}{2}}{\rho_{bs}\left( {r,z} \right)}}}} \sim {2^{\frac{5}{2}}\left( {c/\omega} \right)^{2}{\hat{r}}_{P}^{- \frac{3}{2}}{\int_{{\hat{r}}_{<}}^{{\hat{r}}_{>}}{d{\hat{r}\left( {{\hat{r}}^{2} - 1} \right)}^{- \frac{1}{4}}{n_{1}\left( {L_{\hat{z}}{\omega/c}} \right)}^{\frac{1}{2}}{\langle\rho_{bs}\rangle}}}}},} & (37) \\{\mspace{79mu} {with}} & \; \\{{\mspace{79mu} {{{\langle\rho_{bs}\rangle}(r)} \equiv {\int_{0}^{1}{d\; {{\eta\rho}_{bs}\left( {r,z} \right)}}}}}_{z = {z_{c} - {\eta^{2}L_{\hat{z}}}}},} & (38)\end{matrix}$

where z_(c)−L₂(r)≦z≦z_(c) and r<≦r≦r< are the intervals over which thebifurcation surface intersects the source distribution pattern (see FIG.6). The quantity (ρbs) (r) may be interpreted, at any given r, as aweighted average—over the intersection of the coalescing sheets of thebifurcation surface with the plane z=z_(c)−η²L_({circumflex over (z)})→of the source density ρ.

The right-hand side of (37) decays like

$\left. {{rp}^{- \frac{3}{2}}\mspace{14mu} {as}\mspace{14mu} {rp}}\rightarrow{\infty.} \right.$

The second term in (33) thus dominates the first term in this equation,and so the quantity (∇_(P)A₀)_(out) itself decays like r_(p) ⁻¹ in thefar zone.

B. Time Derivative of the Vector Potential

Inasmuch as the charge density (23) has an unchanging distributionpattern in the (r, {circumflex over (φ)}, z)-frame, the electric currentdensity associated with the moving source distribution pattern we havebeen considering is given by

j(x,t)=rωρ(r,{circumflex over (φ)},z)ê _(φ),   (39)

in which rωê_(φ)=rω[− sin(φ−φ_(P))ê_(rP)+ cos(φ−φ_(P))ê_(φP)] is thevelocity of the element of the source distribution pattern that islocated at (r, {circumflex over (φ)}, z). This current satisfies thecontinuity equation ∂ρ/∂(ct)+∇·j=0 automatically.

In the Lorentz gauge, the retarded vector potential corresponding to(24a) has the form

A(x _(P) ,t _(P))=c ⁻¹ ∫d ³ xdtj(x,t)δ(t _(P) −t−|x−x _(P) |/c)/|x−x_(P)|.   (40)

If we insert (39) in (40) and change the variables of integration from(r, φ, z, t) to (r, φ, z, {circumflex over (φ)}), as in (24), we obtain

A=∫dV{circumflex over (r)}ρ(r, {circumflex over (φ)}, z)G ₂(r, r _(P),{circumflex over (φ)}−{circumflex over (φ)}_(P) , z−z _(P)),   (41)

in which dV=rdrd{circumflex over (φ)} dz, the vector G₂—which plays therole of a Green's function—is given by

$\begin{matrix}{{{G_{2} \equiv {\int_{- \infty}^{+ \infty}{d\; \phi {\hat{e}}_{\phi}{{\delta \left\lbrack {{g(\phi)} - \varphi} \right\rbrack}/{R(\phi)}}}}} = {\sum\limits_{\phi = \phi_{j}}\; {R^{- 1}{{{\partial g}/{\partial\phi}}}^{- 1}{\hat{e}}_{\phi}}}},} & (42)\end{matrix}$

and g and φ_(j)s are the same quantities as those appearing in (17) (seealso FIG. 2).

Because (17), (34) and (42) have the factor |∂_(g)/∂φ|⁻¹ in common, thefunction G₂ has the same singularity structure as those of G₀ and G₁: itdiverges on the bifurcation surface ∂_(g)/∂φ=0 if this surface isapproached from inside, and it is most singular on the cusp curve of thebifurcation surface where in addition ∂²g/∂φ²=0. It is, moreover,described by two different expressions, G₂ ^(in) and G₂ ^(out), insideand outside the bifurcation surface whose asymptotic values in theneighbourhood of the cusp curve have exactly the same functional formsas those found in (18) and (19); the only difference being that p₀ andq₀ in these expressions are replaced by the p₂ and q₂ given in (A37)(see Appendix A).

As in (29), therefore, the time derivative of the vector potential hasthe form ∂A/∂t_(P)=(∂A/∂t_(P))_(in)+(∂A/∂t_(P))_(out) with

(∂A/∂t_(P))_(in,out)≡−ω∫_(V) _(in,out) dV{circumflex over (r)}ρ∂G₂^(in,out)/∂{circumflex over (φ)}_(P)   (43)

when the observation point is such that the bifurcation surfaceintersects the source distribution pattern.

The functions G₂ ^(in,out) depend on {circumflex over (φ)}_(p) and{circumflex over (φ)} in the combination {circumflex over(φ)}−{circumflex over (φ)}_(p) only. We can therefore replace∂/∂{circumflex over (φ)}_(p) in (43) by—∂/∂{circumflex over (φ)} andperform the integration with respect to {circumflex over (φ)} by partsto arrive at

$\begin{matrix}{\mspace{79mu} {{\left( {{\partial\alpha}/{\partial t_{p}}} \right)_{in} = {c{\int_{S}{{drdz}{\hat{r}}^{2}\left\{ {\left\lbrack {\rho \; G_{2}^{in}} \right\rbrack_{\varphi = \varphi_{-}}^{\varphi = \varphi_{+}} - {\int_{\varphi_{-}}^{\varphi_{+}}{d\; \varphi {{\partial\rho}/{\partial\hat{\phi}}}G_{2}^{in}}}} \right\}}}}},}} & (44) \\{\mspace{79mu} {and}} & \; \\{\left( {{\partial\alpha}/{\partial t_{p}}} \right)_{out} = {{- c}{\int_{S}{{drdz}{\hat{r}}^{2}{\left\{ {\left\lbrack {\rho \; G_{2}^{out}} \right\rbrack_{\varphi = \varphi_{-}}^{\varphi = \varphi_{+}} + {\left( {\int_{- }^{\varphi_{-}}{+ \int_{\varphi_{+}}^{+ }}} \right)d\; \varphi {{\partial\rho}/{\partial\hat{\phi}}}G_{2}^{out}}} \right\}.}}}}} & (45)\end{matrix}$

For the same reasons as those given in the paragraphs following (32) and(33), Hadamard's finite part of (∂A/∂t_(p))_(in) consists of the volumeintegral in (44) and is of the order of

${\hat{r}}_{p}^{- \frac{3}{2}}$

[note that according to (A37) and (A42), p₂>>c₁q₂ and p₂/c₁ ²=O(i)]. Thevolume integral in (45), moreover, decays like

r̂_(p)⁻¹,

as does its counterpart in (33).

The part of ∂A/∂t_(p) that decays more slowly than conventionalcontributions to a radiation field is the boundary term in (45). Theasymptotic value of this term is given by an expression similar to thatappearing in (36), except that p₁ and q₁ are replaced by p₂ and q_(2.)Once the quantities and ρ|φ_(±) in the expression in question areapproximated by ρ_(bs) and by (A42), as before, it follows that

$\begin{matrix}{\; {{{\left. \left( {{\partial\alpha}/{\partial t_{p}}} \right)_{out} \right.\sim{- c}}{\int_{S}{{drdz}{{\hat{r}}^{2}\left\lbrack {\rho \; G_{2}^{out}} \right\rbrack}_{\varphi_{-}}^{\varphi_{+}}}}} \sim {{- \frac{4}{3}}c{\int_{S}{{drdz}{\hat{r}}^{2}\rho_{bs}{c_{1}}^{- 1}q_{2}}}} \sim {{- \frac{2^{\frac{3}{2}}}{3}}\left( {c^{2}/\omega} \right){\hat{r}}_{p}^{- 1}{\hat{e}}_{\phi p}{\int_{{\hat{r}}_{<}}^{{\hat{r}}_{>}}{d\hat{r}{{\hat{r}}^{2}\left( {{\hat{r}}^{2} - 1} \right)}^{- \frac{1}{4}}{\int_{{\hat{z}}_{c - {L_{\hat{z}}{\omega/c}}}}^{{\hat{z}}_{c}}{d{\hat{z}\left( {{\hat{z}}_{c} - \hat{z}} \right)}^{- \frac{1}{2}}{\rho_{bs}.}}}}}}}} & (46)\end{matrix}$

This behaves like

${\hat{r}}_{p}^{- \frac{1}{2}}.$

as {circumflex over (r)}_(p)→∞ since the {circumflex over(z)}-quadrature in (46) has the finite value

$2\left( {L_{\hat{z}}{\omega/c}} \right)^{\frac{1}{2}}{\langle\rho_{bs}\rangle}$

in this limit [see (37) et seq.].

Hence, the electric field vector of the radiation

$\begin{matrix}{E = {{{- {\nabla_{p}A_{0}}} - {{\partial\alpha}/{\partial\left( {ct}_{p} \right)}}} \sim {- {c^{- 1}\left( {{\partial\alpha}/{\partial t_{p}}} \right)}_{out}} \sim {\frac{2^{\frac{\tau}{2}}}{3}\left( {c/\omega} \right){\hat{r}}_{p}^{- \frac{1}{2}}{\hat{e}}_{\phi \; p}{\int_{{\hat{r}}_{<}}^{{\hat{r}}_{>}}{d\hat{r}{{\hat{r}}^{2}\left( {{\hat{r}}^{2} - 1} \right)}^{- \frac{1}{4}}\left( {L_{\hat{z}}{\omega/c}} \right)^{\frac{1}{2}}{\langle\rho_{bs}\rangle}}}}}} & (47)\end{matrix}$

itself decays like

$r_{p}^{- \frac{1}{2}}$

in the far zone: as we have already seen in Sec. IV(A), the term ∇_(P)A₀has the conventional rate of decay r_(p) ⁻¹ and so is negligiblerelative to (∂A/∂t_(p))_(out).

C. Curl of the Vector Potential

There are no contributions from the limits of integration towards thecurl of the integral in (41) because ρ vanishes outside a finite volumeand so the integral in this equation extends over all values of (r,{circumflex over (φ)}, z). Hence, differentiation of (41) yields

B=∇ _(P) ×A=B _(in) +B _(out),   (48a)

in which

B_(in,out)≡∫_(V) _(in,out) dV{circumflex over (r)}ρ∇_(P)×G₂ ^(in,out).  (48b)

Operating with ∇_(P)× on the first member of (42) and ignoring the termthat decays like r_(p) ⁻², as in (30), we find that the kernels.∇_(P)×G₂ ^(in) and ∇_(P)×G₂ ^(out) of (48b) are given—in the radiationzone—by the values of

∇_(P) ×G ₂≅(ω/c)∫_(−∞) ^(+∞) dφR ⁻¹δ′(g−φ){circumflex over (n)}×ê _(φ) ,{circumflex over (r)} _(P)>>1,   (49)

for φ inside and outside the interval (φ⁻, φ₊), respectively.[{circumflex over (n)} is the unit vector defined in (31).]

Insertion of (49) in (48) now yields expressions whose {circumflex over(φ)}-quadratures can be evaluated by parts to arrive at

$\begin{matrix}{\mspace{79mu} {{Β_{in} \simeq {\int_{S}{{drdz}{\hat{r}}^{2}\left\{ {{\cdots \mspace{14mu}\left\lbrack {\rho G}_{3}^{in} \right\rbrack}_{\varphi = \varphi_{-}}^{\varphi = \varphi_{+}} + {\int_{\varphi_{-}}^{\varphi_{+}}{d\; \varphi {{\partial\rho}/{\partial\hat{\phi}}}G_{3}^{in}}}} \right\}}}},\mspace{79mu} {{\hat{r}}_{p}1},}} & (50) \\{\mspace{79mu} {and}} & \; \\{{Β_{out} \simeq {\int_{S}{{drdz}{\hat{r}}^{2}\left\{ {\left\lbrack {\rho \; G_{3}^{out}} \right\rbrack_{\varphi = \varphi_{-}}^{\varphi = \varphi_{+}} + {\left( {\int_{- }^{\varphi_{-}}{+ \int_{\varphi_{+}}^{+ }}} \right)d\; \varphi {{\partial\rho}/{\partial\hat{\phi}}}G_{3}^{out}}} \right\}}}},\mspace{79mu} {{\hat{r}}_{p}1},} & (51)\end{matrix}$

where G₃ ^(in) and G₃ ^(out) stand for the values of

$\begin{matrix}{G_{3} = {{\int_{- \infty}^{+ \infty}{d\; \phi \; R^{- 1}{\delta \left( {g - \varphi} \right)}\hat{n} \times {\hat{e}}_{\phi}}} = {\sum\limits_{\phi = \phi_{j}}\; {R^{- 1}{{{\partial g}/{\partial\phi}}}^{- 1}\hat{n} \times {\hat{e}}_{\phi}}}}} & (52)\end{matrix}$

inside and outside the bifurcation surface.

Once again, owing to the presence of the factor |∂_(g)/∂φ|⁻¹ in G₃^(in), the first term in (50) is divergent so that the Hadamard's finitepart of B_(in) consists of the volume integral in this equation, anintegral whose magnitude is of the order of

${\hat{r}}_{p}^{- \frac{3}{2}}$

[see the paragraph containing (35) and note that, according to (A38) and(A44), p₃>>c₁ q₃ and p₃/c₁ ²=O(1)]. The second term in (51) has—likethose in (33) and (45)—the conventional rate of decay {circumflex over(r)}_(P) ⁻¹. Moreover, the surface integral in (51)—which would have hadthe same magnitude as the surface integral in (50) and so would havecancelled out of the expression for B had G₃ ^(in) and G₃ ^(out) matchedsmoothly across the bifurcation surface—decays as slowly as thecorresponding term in (45).

The asymptotic value of G₃ for source points close to the cusp curve ofthe bifurcation surface has been calculated in Appendix A. It followsfrom this value of G₃ and from (51), (52), (A40), (A44) and (A45) that,in the radiation zone,

$\begin{matrix}{Β \sim {\int_{S}{{drdz}{{\hat{r}}^{2}\left\lbrack {\rho \; G_{3}^{out}} \right\rbrack}_{\varphi_{-}}^{\varphi_{+}}}} \sim {\frac{4}{3}{\int_{S}{{drdz}{\hat{r}}^{2}\rho_{bs}c_{1}^{- 1}q_{3}}}} \sim {\frac{2^{\frac{3}{2}}}{3}\left( {c/\omega} \right){\hat{r}}_{P}^{- \frac{1}{2}}{\int_{{\hat{r}}_{<}}^{{\hat{r}}_{>}}{d\hat{r}{{\hat{r}}^{2}\left( {{\hat{r}}^{2} - 1} \right)}^{- \frac{1}{4}}{\int_{{\hat{z}}_{c - {L_{\hat{z}}{\omega/c}}}}^{{\hat{z}}_{c}}{d{\hat{z}\left( {{\hat{z}}_{c} - \hat{z}} \right)}^{- \frac{1}{2}}\rho_{bs}n_{3}}}}}}} & (53)\end{matrix}$

to within the order of the approximation entering (37) and (46).

The far-field version of the radial unit vector defined in (31) assumesthe form

$\begin{matrix}{\lim\limits_{r_{P}\rightarrow\infty}{\hat{n}{_{\varphi = {{\varphi_{c,}\hat{z}} = {\hat{z}}_{c}}}{= {{{\hat{r}}^{- 1}{\hat{e}}_{r_{P}}} - {\left( {1 - {\hat{r}}^{- 2}} \right)^{\frac{1}{2}}{\hat{e}}_{z_{P}}}}}}}} & (54)\end{matrix}$

on the cusp curve of the bifurcation surface [see (12b), (13) and (A27),and note that the position of the observer is here assumed to be suchthat the segment of the cusp curve lying within the source distributionpattern is described by the expression with the plus sign in (12b), asin FIG. 6]. So, n₃ equals {circumflex over (n)}×ê_(φP) in the regime ofvalidity of (53) [see (A45)]. Moreover, {circumflex over (n)} can bereplaced by its far-field value

{circumflex over (n)}≅(r_(P)ê_(r) _(P) +z_(P)ê_(z) _(P) )/R_(P),R_(P)→∞,   (55)

if it is borne in mind that (53) holds true only for an observer thecusp curve of whose bifurcation surface intersects the sourcedistribution pattern.

Once n₃ in (53) is approximated by {circumflex over (n)}×ê_(φP) and theresulting {circumflex over (z)}-quadrature is expressed in terms of

ρbs

[see (38)], this equation reduces to

B˜{circumflex over (n)}×E,   (56),

where E is the electric field vector earlier found in (47). Equations(47) and (56) jointly describe a radiation field whose polarizationvector lies along the direction of motion of the source distributionpattern, ê_(φP).

Note that there has been no contribution toward the values of E and Bfrom inside the bifurcation surface. These quantities have arisen in theabove calculation solely from the jump discontinuities in the values ofthe Green's functions G₁ ^(out), G₂ ^(out) and G₃ ^(out) across thecoalescing sheets of the bifurcation surface. We would have obtained thesame results had we simply excised the vanishingly small volumelim_(rp)→∞V_(in) from the domains of integration in (29), (43) and (48).

Note also that the way in which the familiar relation (56) has emergedfrom the present analysis is altogether different from that in which itappears in conventional radiation theory. Essential though it is to thephysical requirement that the directions of propagation of the waves andof their energy should be the same, (56) expresses a relationshipbetween fields that are here given by non-spherically decaying surfaceintegrals rather than by the conventional volume integrals that decaylike r_(p) ⁻¹.

V. A Physical Description of the Emission Process

Expressions (47) and (56) for the electric and magnetic fields of theradiation that arises from a charge-current density with the components(23) and (39) imply the following Poynting vector:

$\begin{matrix}{S\text{∼}\frac{2^{5}}{3^{2}}\pi^{- 1}{c\left( {c/\omega} \right)}^{2}{{rp}^{- 1}\left\lbrack {\int_{{\hat{r}}_{<}}^{{\hat{r}}_{>}}{d\hat{r}{{\hat{r}}^{2}\left( {{\hat{r}}^{2} - 1} \right)}^{- \frac{1}{4}}\left( {L_{\hat{z}}{\omega/c}} \right)^{\frac{1}{2}}{\langle\rho_{bs}\rangle}}} \right\rbrack}^{2}{\hat{n}.}} & (57)\end{matrix}$

In contrast, the magnitude of the Poynting vector for the coherentcyclotron radiation that would be generated by a macroscopic lump ofcharge, if it moved subluminally with a centripetal acceleration cw isof the order of (

ρ

L³)²ω²/(cR_(P) ²) according to the Larmor formula, where L₃ representsthe volume of the source distribution pattern and

ρ

its average charge density. The intensity of the present emission istherefore greater than that of even a coherent conventional radiation bya factor of the order of (L_({circumflex over (z)})/L)(Lω/c)⁻⁴(R_(P)/L),a factor that ranges from 10¹⁶ to 10³⁰ in the case of pulsars forinstance.

The reason this ratio has so large a value in the far field (R_(P)/L>>1)is that the radiative characteristics of a volume-distributed sourcepattern which moves faster than the waves it emits are radicallydifferent from those of a corresponding source that moves more slowlythan the waves it emits. There are elements of the distirbution patternof the source in the former case that approach the observer along theradiation direction with the wave speed at the retarded time. These lieon the intersection of the source distribution pattern with what we havehere called the bifurcation surface of the observer (see FIGS. 5 and 6):a surface issuing from the position of the observer which has the sameshape as the envelope of the wave fronts emanating from an element ofthe source distribution pattern (FIGS. 1 and 3A) but which spiralsaround the rotation axis in the opposite direction to this envelope andresides in the space of source points instead of the space ofobservation points.

The elements of the source distribution pattern inside the bifurcationsurface of an observer make their contributions towards the observedfield at three distinct instants of the retarded time. The values of twoof these retarded times coincide for an interior element of the sourcedistribution pattern that lies next to the bifurcation surface. Thislimiting value of the coincident retarded times represents the instantat which the component of the velocity of the element in question of thesource distribution pattern equals the wave speed c in the direction ofthe observer. The third retarded time at which an element of the sourcedistribution pattern adjacent to—just inside—the bifurcation surfacemakes a contribution is the same as the single retarded time at whichits neighbouring element of the source distribution pattern just outsidethe bifurcation surface makes its contribution towards the observedfield. (The elements of the source distribution pattern outside thebifurcation surface make their contributions at only a single instant ofthe retarded time).

At the instant marked by this third value of the retarded time, the twoneighbouring elements of the source distribution pattern—just interiorand just exterior to the bifurcation surface—have the same velocity, buta velocity whose component along the radiation direction is differentfrom c. The velocities of these two neighbouring elements are, ofcourse, equal at any time. However, at the time they approach theobserver with the wave speed, the element inside the bifurcation surfacemakes a contribution towards the observed field while the one outsidethis surface does not: the observer is located just inside the envelopeof the wave fronts that emanate from the interior element of the sourcedistribution pattern but just outside the envelope of the wave frontsthat emanate from the exterior one. Thus, the constructive interferenceof the waves that are emitted by the element of the source distributionpattern just outside the bifurcation surface takes place along a causticwhich at no point propagates past the observer at the conical apex ofthe bifurcation surface in question.

On the other hand, the radiation effectiveness of an element of thedistribution pattern of the source which approaches the observer withthe wave speed at the retarded time is much greater than that of aneighbouring element the component of whose velocity along the radiationdirection is subliminal or superluminal at this time. This is becausethe piling up of the emitted wave fronts along the line joining thesource and the observer makes the ratio of emission to reception timeintervals for the contributions of the luminally moving elements of thesource distribution pattern by many orders of magnitude greater thanthat for the contributions of any other elements. As a result, theradiation effectiveness of the various constituent elements of thesource distribution pattern (i.e. the Green's function for the emissionprocess) undergoes a discontinuity across the boundary set by thebifurcation surface of the observer.

The integral representing the superposition of the contributions of thevarious volume elements of the source distribution pattern to thepotential thus entails a discontinuous integrand. When this volumeintegral is differentiated to obtain the field, the discontinuity inquestion gives rise to a boundary contribution in the form of a surfaceintegral over its locus. This integral receives contributions fromopposite faces of each sheet of the bifurcation surface which do notcancel one another. Moreover, the contributions arising from theexterior faces of the two sheets of the bifurcation surface do not havethe same value even in the limit R_(p)→∞. where this surface isinfinitely large and so its two sheets are—throughout a localized sourcedistribution pattern that intersects the cusp—coalescent. Thus theresulting expression for the field in the radiation zone entails asurface integral such as that which would arise if the sourcedistribution pattern were two-dimensional, i.e. if the sourcedistribution pattern were concentrated into an infinitely thin sheetthat coincided with the intersection of the coalescing sheets of thebifurcation surface with the source distribution pattern.

For a two-dimensional source distribution pattern of this type—whetherit be real or a virtual one whose field is described by a surfaceintegral—the near zone (the Fresnel regime) of the radiation can extendto infinity, so that the amplitudes of the emitted waves are notnecessarily subject to the spherical spreading that normally occurs inthe far zone (the Fraunhofer regime). The Fresnel distance which marksthe boundary between these two zones is given by R_(F)˜L_(⊥) ²/L_(∥), inwhich L_(⊥) and L_(∥) are the dimensions of the source distributionpattern perpendicular and parallel to the radiation direction. If thedistribution pattern of the source is distributed over a surface and sohas a dimension L_(∥) that is vanishingly small, therefore, the Fresneldistance R_(F) tends to infinity.

In the present case, the surface integral which arises from thediscontinuity in the radiation effectiveness of the source elementsacross the bifurcation surface has an integrand that is in turn singularon the cusp curve of this surface. This has to do with the fact that theelements the source distribution pattern on the cusp curve of thebifurcation surface approach the observer along the radiation directionnot only with the wave speed but also with zero acceleration. The ratioof the emission to reception time intervals for the signals generated bythese elements is by several orders of magnitude greater even than thatfor the elements on the bifurcation surface. When the contributions ofthese elements are included in the surface integral in question, i.e.when the observation point is such that the cusp curve of thebifurcation surface intersects the source distribution pattern (as shownin FIG. 6), the value of the resulting improper integral turns out tohave the dependence R_(p) ^(−1/2), rather than R_(p) ⁻¹, on the distanceR_(p) of the observer from the source distribution pattern.

This non-spherically decaying component of the radiation is in additionto the conventional component that is concurrently generated by theremaining volume elements of the source distribution pattern. It isdetectable only at those observation points the cusp curves of whosebifurcation surfaces intersect the source distribution pattern. Itappears, therefore, as a spiral-shaped wave packet with the sameazimuthal width as the {circumflex over (φ)}-extent of the sourcedistribution pattern. For a source distribution pattern whosesuperluminal portion extends from {circumflex over (r)}=1 to {circumflexover (r)}={circumflex over (r)}_(>)>1, this wave packet is detectable—byan observer at infinity—within the angles

${{\frac{1}{2}\pi} - {\arccos \; {\hat{r}}_{>}^{- 1}}} \leq {\theta \; p} \leq {{\frac{1}{2}\pi} + {\arccos {\hat{r}}_{>}^{- 1}}}$

from the rotation axis: projection (12b) of the cusp curve of thebifurcation surface onto the (r, z)-plane reduces to

${\cot \; \theta \; p} = \left( {{\hat{r}}^{2} - 1} \right)^{\frac{1}{2}}$

in the limit R_(P)→∞, where θ_(P)≡arctan(r_(P)/z_(P)) [also see (54)].

Because it comprises a collection of the spiralling cusps of theenvelopes of the wave fronts that are emitted by various elements of thesource distribution pattern, this wave packet has a cross section withthe plane of rotation whose extent and shape match those of the sourcedistribution pattern. It is a diffraction-free propagating causticthat—when detected by a far-field observer—would appear as a pulse ofduration Δ{circumflex over (φ)}/ω, where Δ{circumflex over (φ)} is theazimuthal extent of the source distribution pattern.

Note that the waves that interfere constructively to form each cusp, andhence the observed pulse, are different at different observation times:the constituent waves propagate in the radiation direction {circumflexover (n)} with the speed c, whereas the propagating caustic that isobserved, i.e. the segment of the cusp curve that passes through theobservation point at the observation time, propagates in the azimuthaldirection ê_(φP) with the phase speed r_(pw).

The fact that the intensity of the pulse decays more slowly thanpredicted by the inverse square law is not therefore incompatible withthe conservation of energy, for it is not the same wave packet that isobserved at different distances from the source distribution pattern:the wave packet in question is constantly dispersed and re-constructedout of other waves. The cusp curve of the envelope of the wavefrontsemanating from an infinitely long-lived source distribution pattern isdetectable in the radiation zone not because any segment of this curvecan be identified with a caustic that has formed at the source and hassubsequently travelled as an isolated wavepacket to the radiation zone,but because certain set of waves superpose coherently only at infinity.

Relative phases of the set of waves that are emitted during a limitedtime interval is such that these waves do not, in general, interfereconstructively to form a cusped envelope until they have propagated somedistance away from the source distribution pattern. The period in whichthis set of waves has a cusped envelope and so is detectable as aperiodic train of non-spherically decaying pulses, would of course havea limited duration if the source distribution pattern is short-lived.

Thus, pulses of focused waves may be generated by the present emissionprocess which not only are stronger in the far field than any previouslystudied class of signals, but which can in addition be beamed at only aselect set of observers for a limited interval of time.

VI. Description of Examples of the Apparatus

An apparatus can be designed for generating such pulses, in accordancewith the above theory, which basically entails the simple componentsshown in FIGS. 7A and 7B.

Referring to the example of FIG. 7A, a linear dielectric rod 1 oflength/is provided with an array of electrodes 2, 3 arranged oppositeone another along its length with n/l electrodes per unit length. Inuse, a voltage potential is applied across the dielectric rod 1 by theelectrodes 2, 3, with each pair of electrodes 2, 3, in the array beingactivated in turn to generate a polarisation region with the fronts 5.By rapid application and removal of a potential voltage to electrodes 2,3, the distribution pattern of this polarised region can be set inaccelerated motion with a superluminal velocity. Creating a voltageacross a pair of electrodes polarises the material in the rod betweenthe electrodes. The electrodes can be controlled independently, so thatthe distribution pattern of polarisation of the rod as a function oflength along the rod is controlled.

By varying the voltage across the electrode pairs as a function of time,this polarisation pattern is set in motion. For example, neighbouringelectrode pairs can be turned on with a time interval of Δt betweenthem, starting from one end of the rod. Thus, at a snapshot in time,part of the rod is polarised (that part lying between electrode pairswith a voltage across them) and part of it is not polarised (that partlying between electrode pairs without a voltage across them). Theseregions are separated by “polarisation fronts” which move with a speedof l/(nΔt). With suitable choices of n and Δt the polarisation frontscan be made to move at any speed (including speeds faster than the speedof light in vacuo). The polarisation fronts can be accelerated throughthe speed of light by changing Δt with time.

High-frequency radiation may be generated by modulating the amplitude ofthe resulting polarisation current with a frequency Ω that exceeds a/c,where a is the acceleration of the source distribution pattern. Thespectrum of the spherically decaying component of the radiation wouldthen extend to frequencies that would be by a factor of the order of(cΩ/a)² higher than Ω. The required modulation may be achieved byvarying the amplitudes of the voltages that are applied across variouselectrode pairs all in phase.

FIG. 7B shows another example of the invention, the one analysed above.In this example, the dielectric rod is formed in the shape of a ring.FIG. 7B is a plan view showing electrodes 2, and has electrodes 3disposed below the rod 1. For a ring of radius r and a polarisationpattern that moves around the ring with an angular frequency ω, thevelocity of the charged region is rω. In this example, rω is greaterthan the speed of light c so that the moving polarisation pattern emitsthe radiation described with reference to FIGS. 1 to 6. FIGS. 3B and 3Cdepict representative three dimensional plots of the radiation patternof the entire source of FIG. 7B at a frequency of 2.4 GHz and a phasedifference between adjacent electrodes between 15 degrees and 5 degreesrespectively. An azimuthal or radial polarisation current may beproduced by displacing the plates of each electrode pair relative to oneanother.

The voltages across neighbouring electrode pairs have the same timedependence (their period is 2π/ω) but, as in the rectilinear case, thereis a time difference of Δt between them. The polarisation distributionpattern must move coherently around the ring, i.e. must move rigidlywith an unchanging shape; this would be the case if nΔt=2πN/ω, where nis the number of electrodes around the ring and N an integer. Within theconfines of this condition, the time dependence of the voltage acrosseach pair of electrodes can be chosen at will. The exact form of theadopted time dependence would allow, for example, the generation ofharmonic content and structure in the source distribution pattern. As inthe rectilinear case, modulation of the amplitude of this sourcedistribution pattern at a frequency Ω would result in a radiation whosespectrum would contain frequencies of the order of (Ω/ω)²Ω.

The electrodes are driven by an array of similar oscillators, an arrayin which the phase difference between successive oscillators has a fixedvalue. There are several ways of implementing this:

a single oscillator may be used to drive each electrode throughprogressively longer delay lines;

each electrode pair may be driven by an individual oscillator in anarray of phase-locked oscillators; or

the electrode pairs may be connected to points around a circle of radiusr which lies within—and is coplanar with—an annular waveguide, awaveguide whose normal modes include an electromagnetic wave train thatpropagates longitudinally around the circle with an angular frequencyω>c/r.

For a dielectric rod in the shape of a ring of diameter 1 m, oscillatorsoperating at a frequency of 100 MHz would generate a superluminallymoving polarisation distribution pattern. The required oscillatorfrequencies are easily obtainable using standard laboratory equipment,and any material with an appreciable polarizability at MHz frequencieswould do for the medium. If the amplitude of the resulting polarisationcurrent is in addition modulated at 1 GHz, then the device would radiateat ˜100 GHz. The efficiency of this emission process is expected to beas high as a few percent.

With oscillators operating at frequencies of 1 GHz (also available), thesize of the device would be about 10 cm across; applications demandingportability are therefore viable.

VII. Applications A. Medical and Biomedical Applications

The present invention may be exploited to generate waves which do notform themselves into a focused pulse until they arrive at their intendeddestination and which subsequently remain in focus only for anadjustable interval of time, a property that allows for applications invarious areas of medical practice and biomedical research.

Examples of its use in therapeutic medicine are: (i) the selectiveirradiation of deep tumours whilst sparing surrounding normal tissue,and (ii) the radiation pressure or thermocautery removal of thromboticand embolic vascular lesions that may result from abnormalities in bloodclotting without invasive surgery. Examples of its use in diagnosticmedicine are absorption spectroscopy (focusing a broadband pulse withina tissue some frequencies of which would be absorbed) andthree-dimensional tomography (mapping specifiable regions of interestwithin the body to high levels of resolution). In biomedical research,it provides a more powerful alternative to confocal scanning microscopy;with a single aerial being used as an X-ray source for imaging purposes.

An example of an apparatus required for generating the pulses inquestion is that shown in FIG. 7A. It consists of a linear dielectricrod, an array of electrode pairs positioned opposite to each other alongthe rod, and the means for applying a voltage to the electrodessequentially at a rate sufficient to induce a polarization current whosedistribution pattern moves along the rod with a constant acceleration atspeeds exceeding the speed of light in vacuo.

The envelope of the wave fronts emanating from a volume element of thesuperluminally moving distribution pattern thus produced is shown inFIG. 8. It consists of a two-sheeted closed surface when the duration ofthe source includes the instant at which the distribution pattern of thesource becomes superluminal. The two sheets of this envelope are tangentto one another and form a cusp along an expanding circle. If the sourcedistribution pattern has a limited duration, the envelope in question iscorrespondingly limited [as in FIG. 9D] to only a truncated section ofthe surface shown in FIG. 8.

The snapshots in FIGS. 9A-9F trace the evolution in time of the relativepositions of a particular set of wave fronts that are emitted during ashort time interval. They include times at which the envelope has notyet developed a cusp [9A and 9B], has a cusp [9C-9E], and has alreadylost its cusp 9F.

A source distribution pattern with the life span 0<t<T gives rise to acaustic, i.e. to a set of tangential wave fronts with a cusped envelope,only during the following finite interval of observation time:

M(M ²−1)l/c≦t _(P) ≦M[M ²(1+aT/u)³−1]l/c,   (58)

where M≡u/c and l≡c²/a with u, c, and a standing for the speed of thedistribution pattern of the source at t=0, the wave speed, and theconstant acceleration of the distribution pattern of the source,respectively. For a T/u<<1, therefore, the duration of the caustic,3M²T, is proportional to that of the source distribution pattern.

Moreover, a cusped envelope begins to form in the case of a short-livedsource distribution pattern only after the waves have propagated afinite distance away from the source. The distance of the caustic fromthe position of the source distribution pattern at the retarded time isgiven by

$\begin{matrix}{{{\overset{\sim}{R}p} = {\beta \; {p^{\frac{1}{3}}\left( {{\beta \; p^{\frac{2}{3}}} - 1} \right)}l}},} & (59)\end{matrix}$

where β_(P)≡(u+at_(P))/c and t_(p) is the observation time. Thisdistance can be long even when the duration of the source distributionpattern is short because there is no upper limit on the value of thelength l(≡c²/a) that enters (58) and (59):/tends to infinity for a→0 andis as large as 10¹⁸ cm when a equals the acceleration of gravity. ThusR_(P) can be rendered arbitrarily large, by a suitable choice of theparameter I, without requiring either the duration of the source (T) orthe retarded value

$\left( {\beta \; p^{\frac{1}{3}}c} \right)$

of the speed of the source distribution pattern to be correspondinglylarge.

This means that, when either M or I is large, the waves emitted by ashort-lived source do not focus to such an extent as to form a cuspedenvelope until they have travelled a long distance away from the source.The period during which they then do so can be controlled by adjustingthe parameters M and T.

The collection of the cusp curves of the envelopes that are associatedwith various elements of the distribution pattern of the sourceconstitutes a ring-shaped wave packet. This wave packet is interceptedonly by those observers who are located, during its life time (58), onits trajectory

$\begin{matrix}{{\xi = \left( {{\beta \; p^{\frac{2}{3}}} - 1} \right)^{\frac{3}{2}}},{\zeta = {{\frac{1}{2}\beta \; p^{2}} - {\frac{3}{2}\beta \; p^{\frac{3}{2}}} + 1}},} & (60)\end{matrix}$

where ξ represents the distance (in units of I) of the observer from therectilinear path of the source, say the z-axis, and ζ stands for thedifference between the Lagrangian coordinates

$\overset{\sim}{z} = {z - {ut} - {\frac{1}{2}{at}^{2}}}$

of the source point and

${\overset{\sim}{z}}_{P} = {z_{P} - {utp} - {\frac{1}{2}{atp}^{2}}}$

of the observation point.

It is possible to limit the spatial extent of the wave packet embodyingthe large-amplitude pulse by enclosing the path of the sourcedistribution pattern within an opaque cylindrical surface which has anarrow slit parallel to its axis, a slit acting as an aperture thatwould only allow an arc of the ring-shaped wave packet to propagate tothe far field. The volume occupied by the resulting wave packet couldthen be chosen at will by adjusting the width of the aperture and thelongitudinal extent of the source distribution pattern.

B. Compact Sources of Intense Broadband Radiation

In the near zone, the radiation that is generated by the invention canbe arranged to have many features in common with synchrotron radiation.Most experiments presently carried out at large-scale synchrotronfacilities could potentially be performed by means of a polarizationsynchrotron, i.e. the compact device described in Sec. VI. This devicehas applications, as a source of intense broadband radiation, in manyscientific and industrial areas, e.g. in spectroscopy, in semiconductorlithography at very fine length scales, and in silicon chip manufactureinvolving UV techniques.

The spectrum of the radiation generated in a polarization synchrotronextends to frequencies that are by a factor of the order of (cΩ/a)²higher than the characteristic frequency Ω of the fluctuations of thesource distribution pattern itself (c and a are the speed of light andthe acceleration of the source distribution pattern, respectively). Fora polarizable medium consisting of a 1 m arc of a circular rod whosediameter is ˜10 m [see FIG. 7B], superluminal source distributionpattern motion is achieved by an applied voltage that oscillates withthe frequency ˜10 MHz. If the amplitude of the resulting polarizationcurrent is in addition modulated at ˜500 MHz, then the device wouldradiate at ˜1 THz.

In the case of the source distribution pattern elements that approachthe observer with the wave speed and zero acceleration, the interval ofretarded time δt during which a set of waves are emitted issignificantly longer than the interval of observation time δt_(P) duringwhich the same set of waves are received.

For a rectilinearly moving superluminal source distribution pattern, theratio δt/δt_(P) is given by

${2^{\frac{1}{3}}\left( {{u^{2}/c^{2}} - 1} \right)^{\frac{1}{3}}\left( {a\; \delta \; {{tp}/c}} \right)^{- \frac{2}{3}}},$

where u is the retarded speed of the source distribution pattern and aits constant acceleration. This ratio increases without bound as aapproaches zero. Regardless of what the characteristic frequency of thetemporal fluctuations of the source may be, therefore, it is possible topush the upper bound to the spectrum of the emitted radiation toarbitrarily high frequencies by making the acceleration a small. [Notethat the emission process described here remains different from theĈerenkov process, in which a exactly equals zero, even in the limita→0.]

The relationship between δt and δt_(P) is

${\delta \; {tp}} \simeq {\frac{1}{6}{\omega^{2}\left( {\delta \; t} \right)}^{3}}$

if the source distribution pattern moves circularly with the angularfrequency w. Thus the spectrum of the spherically decaying part of theradiation that is generated by a source with an accelerated superluminaldistribution pattern extends to frequencies which are by a factor of theorder of (cΩ/a)² or (Ω/ω)² higher than the characteristic frequency Ω ofthe modulations of the amplitude of the source distribution pattern.

C. Long-Range and High-Bandwidth Telecommunications

There are at present no known antennas in which the emitting electriccurrent is both volume distributed and has the time dependence of atravelling wave with an accelerated superluminal motion. A travellingwave antenna of this type, designed on the basis of the principlesunderlying the present invention, generates focused pulses that not onlyare stronger in the far field than any previously studied class ofsignals, but can in addition be beamed at only a select set of observersfor a limited interval of time: the constituent waves whose constructiveinterference gives rise to the propagating wave packet embodying a givenpulse come into focus (develop a cusped envelope or a caustic) only longafter they have emanated from the source and then only for a finiteperiod (FIGS. 9A-9F).

The intensity of the waves generated by this novel type of antenna decaymuch more slowly over distance than that of conventional radio or lightsignals. In the case of conventional sources, including lasers, if thetransmitter (source) to receiver (destination) distance doubles, thepower of the signal is reduced by a factor of four. With the presentinvention, the same doubling of distance only halves the availablesignal. Thus the power required to send a radio signal from the Earth tothe Moon by the present transmitter would be 100 million times smallerthan that which is needed in the case of a conventional antenna.

The emission mechanism in question can therefore be used to conveytelephonic, visual and other electronic data over very long distanceswithout significant attenuation. In the case of ground-to-satellitecommunications, the power required to beam a signal would be greatlyreduced, implying that either far fewer satellites would be required forthe same bandwidth or each satellite could handle a much wider range ofsignals for the same power output.

D. Hand-Held Communication Devices

A combined effect of the slow decay rate and the beaming of the newradiation is that a network of suitably constructed antennae couldexpand the useable spectrum of terrestrial electromagnetic broadcasts bya factor of a thousand or more, thus dispensing with the need for cableor optical fibre for high-bandwidth communications.

The evolution of the Internet, real-time television conferencing andrelated information-intense communication media means that there is agrowing demand for cheap high-bandwidth aerials. Highly compact aerialsfor hand-held portable phones and/or television/Internet connectionsbased on the present invention can handle, not only much longertransmitter-to-receiver distances than those currently available incellular phone systems, but also much higher bandwidth.

Far fewer ground based aerial structures are required to obtain the samearea coverage. Because there would be no cross-talk between any pairs oftransmitter and receiver, the effective bandwidth of free space could beincreased many thousand-fold, thus allowing, say, for video transmissionbetween hand-held units.

Appendix A: Asymptotic Expansion of the Green's Functions

In this Appendix, we calculate the leading terms in the asymptoticexpansions of the integrals (16), (34), (42) and (52) for small φ₊-φ⁻,i.e. for points close to the cusp curve (12) of the bifurcation surface(or of the envelope of the wavefronts). The method—originally due toChester et al. (Proc. Camb. Phil. Soc., 54, 599, 1957)—which we use is astandard one that has been specifically developed for the evaluation ofradiation integrals involving caustics (see Ludwig, Comm. Pure Appl.Maths, 19, 215, 1966). The integrals evaluated below all have a phasefunction g(φ) whose extrema (φ=φ_(±)) coalesce at the caustic (12).

As long as the observation point does not coincide with the sourcepoint, the function g(φ) is analytic and the following transformation ofthe integration variables in (16) is permissible:

$\begin{matrix}{{{g(\phi)} = {{\frac{1}{3}v^{3}} - {c_{1}^{2}v} + c_{2}}},} & ({A1})\end{matrix}$

where v is the new variable of integration and the coefficients

$\begin{matrix}{c_{1} \equiv {\left( \frac{3}{4} \right)^{\frac{1}{3}}\left( {\varphi_{+} - \varphi_{-}} \right)^{\frac{1}{3}}\mspace{14mu} {and}{\mspace{11mu} \;}c_{2}} \equiv {\frac{1}{2}\left( {\varphi_{+} + \varphi_{-}} \right)}} & ({A2})\end{matrix}$

are chosen such that the values of the two functions on opposite sidesof (A1) coincide at their extrema. Thus an alternative exact expressionfor G₀ is

$\begin{matrix}{{G_{0} = {\int_{- \infty}^{+ \infty}{{{dvf}_{0}(v)}{\delta \left( {{\frac{1}{3}v^{3}} - {c_{1}^{2}v} + c_{2} - \varphi} \right)}}}},} & ({A3})\end{matrix}$

in which

f₀(v)≡R⁻¹dφ/dv.   (A4)

Close to the cusp curve (12), at which c₁ vanishes and the extrema v=±c₁of the above cubic function are coincident, f₀(v) may be approximated byp₀+q₀v, with

$\begin{matrix}{{p_{0} = {\frac{1}{2}\left( {f_{0}{_{v = c_{1}}{+ f_{0}}}_{v = {- c_{1}}}} \right)}},} & ({A5}) \\{q_{0} = {\frac{1}{2}{{c_{1}^{- 1}\left( {f_{0}{_{v = c_{1}}{- f_{0}}}_{v = {- c_{1}}}} \right)}.}}} & ({A6})\end{matrix}$

The resulting expression

$\begin{matrix}{G_{0}\text{∼}{\int_{- \infty}^{+ \infty}{{{dv}\left( {p_{0} + {q_{0}v}} \right)}{\delta \left( {{\frac{1}{3}v^{3}} - {c_{1}^{2}v} + c_{2} - \varphi} \right)}}}} & ({A7})\end{matrix}$

will then constitute, according to the general theory, the leading termin the asymptotic expansion of G₀ for small c₁.

To evaluate the integral in (A7), we need to know the roots of the cubicequation that follows from the vanishing of the argument of the Diracdelta function in this expression. Depending on whether the observationpoint is located inside or outside the bifurcation surface (theenvelope), the roots of

$\begin{matrix}{{{\frac{1}{3}\nu^{3}} - {c_{1}^{2}\nu} + c_{2}} = 0} & ({A8})\end{matrix}$

are given by

$\begin{matrix}{{\nu = {2\; c_{1}{\cos \left( {{\frac{2}{3}n\; \pi} + {\frac{1}{3}\arccos \; \chi}} \right)}}},\mspace{14mu} {{\chi } < 1},} & \left( {{A9}\; a} \right)\end{matrix}$

for n=0, 1 and 2, or by

$\begin{matrix}{{\nu = {2\; c_{1}{{sgn}(\chi)}{\cosh \left( {\frac{1}{3}{arc}\; \cosh {\chi }} \right)}}},\mspace{11mu} {{\chi } > 1},} & \left( {{A9}\; b} \right)\end{matrix}$

respectively, where:

$\begin{matrix}{{\chi \equiv {\left\lbrack {\varphi - {\frac{1}{2}\left( {\varphi_{+} + \varphi_{-}} \right)}} \right\rbrack/\left\lbrack {\frac{1}{2}\left( {\varphi_{+} - \varphi_{-}} \right)} \right\rbrack}} = {\frac{3}{2}{\left( {\varphi - c_{2}} \right)/{c_{1}^{3}.}}}} & ({A10})\end{matrix}$

Note that X equals +1 on the sheet φ=φ₊ of the bifurcation surface (theenvelope) and −1 on φ=φ⁻.

The integral in (A7), therefore, has the following value when theobservation point lies inside the bifurcation surface (the envelope):

$\begin{matrix}{{{\int_{- \infty}^{+ \infty}{d\; {{\nu\delta}\left( {{\frac{1}{3}\nu^{3}} - {c_{1}^{2}\nu} + c_{2}} \right)}}} = {\sum\limits_{n = 0}^{2}\; {c_{1}^{- 2}{{{4\; {\cos^{2}\left( {{\frac{2}{3}n\; \pi} + {\frac{1}{3}\arccos \; \chi}} \right)}} - 1}}^{- 1}}}},\mspace{20mu} {{\chi } < 1.}} & ({A11})\end{matrix}$

Using the trignometric identity 4 cos² α−1=sin 3α/sin α, we can writethis as

$\begin{matrix}{{{{\int_{- \infty}^{+ \infty}{d\; {\nu\delta}\left( {{\frac{1}{3}\nu^{3}} - {c_{1}^{2}\nu} + c_{2}} \right)}} = {{{c_{1}^{- 2}\left( {1 - \chi^{2}} \right)}^{- \frac{1}{2}}{\sum\limits_{n = 0}^{2}\; {{\sin \left( {{\frac{2}{3}n\; \pi} + {\frac{1}{3}\arccos \; \chi}} \right)}}}} = {2\; {c_{1}^{- 2}\left( {1 - \chi^{2}} \right)}^{- \frac{1}{2}}{\cos \left( {\frac{1}{3}\arcsin \; \chi} \right)}}}},\mspace{20mu} {{\chi } < 1},}\;} & ({A12})\end{matrix}$

in which we have evaluated the sum by adding the sine functions two at atime.

When the observation point lies outside the bifurcation surface (theenvelope), the above integral receives a contribution only from thesingle value v given in (A9b) and we obtain

$\begin{matrix}{{{{\int_{- \infty}^{+ \infty}{d\; {\nu\delta}\left( {{\frac{1}{3}\nu^{3}} - {c_{1}^{2}\nu} + c_{2}} \right)}} = {{c_{1}^{- 2}\left( {\chi^{2} - 1} \right)}^{- \frac{1}{2}}{\sinh \left( {\frac{1}{3}{arc}\; \cosh \; {\chi }} \right)}}},\mspace{20mu} {{\chi } > 1},}\;} & ({A13})\end{matrix}$

where this time we have used the identity 4 cos h² α−1=sin h 3α/sin hα.

The second part of the integral in (A7) can be evaluated in exactly thesame way. It has the value

$\begin{matrix}{{{{\int_{- \infty}^{+ \infty}{d\; {{\nu\nu\delta}\left( {{\frac{1}{3}\nu^{3}} - {c_{1}^{2}\nu} + c_{2}} \right)}}} = {{2\; {c_{1}^{- 1}\left( {1 - \chi^{2}} \right)}^{- \frac{1}{2}}{\sum\limits_{n = 0}^{2}\; {{{\sin \left( {{\frac{2}{3}n\; \pi} + {\frac{1}{3}\arccos \; \chi}} \right)}} \times {\cos \left( {{\frac{2}{3}n\; \pi} + {\frac{1}{3}\arccos \; \chi}} \right)}}}} = {{- 2}\; {c_{1}^{- 1}\left( {1 - \chi^{2}} \right)}^{- \frac{1}{2}}{\sin \left( {\frac{2}{3}\arcsin \; \chi} \right)}}}},\mspace{20mu} {{\chi } < 1},}\;} & ({A14})\end{matrix}$

when the observation point lies inside the bifurcation surface (theenvelope), and the value

$\begin{matrix}{{{{\int_{- \infty}^{+ \infty}{d\; {{\nu\nu\delta}\left( {{\frac{1}{3}\nu^{3}} - {c_{1}^{2}\nu} + c_{2}} \right)}}} = {{c_{1}^{- 1}\left( {\chi^{2} - 1} \right)}^{- \frac{1}{2}}{{sgn}(\chi)}{\sinh \left( {\frac{2}{3}{arc}\; \cosh \; {\chi }} \right)}}},\mspace{20mu} {{\chi } > 1},}\;} & ({A15})\end{matrix}$

when the observation point lies outside the bifurcation surface (theenvelope).

Inserting (A12)-(A15) in (A7), and denoting the values of G₀ inside andoutside the bifurcation surface (the envelope) by G₀ ^(in) and G₀^(out), we obtain

$\begin{matrix}{{G_{0}^{in} \sim {2\; {{c_{1}^{- 2}\left( {1 - \chi^{2}} \right)}^{- \frac{1}{2}}\left\lbrack {{p_{0}{\cos \left( {\frac{1}{3}\arcsin \; \chi} \right)}} - {c_{1}q_{0}{\sin \left( {\frac{2}{3}\arcsin \; \chi} \right)}}} \right\rbrack}}},\mspace{20mu} {{\chi } < 1},\; {and}} & ({A16}) \\{{{G_{0}^{out} \sim {{c_{1}^{- 2}\left( {\chi^{2} - 1} \right)}^{- \frac{1}{2}}\left\lbrack {{p_{0}{\sinh \left( {\frac{1}{3}{arc}\; \cosh \; {\chi }} \right)}} + {c_{1}q_{0}{{sgn}(\chi)}{\sinh \left( {\frac{2}{3}{arc}\; \cosh \; {\chi }} \right)}}} \right\rbrack}},\mspace{20mu} {{\chi } > 1},}\mspace{31mu}} & ({A17})\end{matrix}$

for the leading terms in the asymptotic approximation to G₀ for smallc₁.

The function f₀(v) in terms of which the coefficients p₀ and q₀ aredefined is indeterminate at v=c₁ and v=−c₁: differentiation of (A1)yields dφ/dυ=(υ²−c₁ ²)/(∂_(g)/∂φ) the zeros of whose denominator at φ=φ⁻and φ=φ₊ respectively coincide with those of its numerator at v=+c₁ andv=−c₁. This indeterminacy can be removed by means of l′Hopital's rule bynoting that

$\begin{matrix}{{\left. \frac{d\; \phi}{d\; \nu} \right|_{\nu = {\pm c_{1}}} = {\left. \frac{\nu^{2} - c_{1}^{2}}{\frac{\partial g}{\partial\phi}} \right|_{\nu = {\pm c_{1}}} = \left. \frac{2\nu}{\left( \frac{\partial^{2}g}{\partial\phi^{2}} \right)\left( \frac{d\; \phi}{d\; \nu} \right)} \right|_{\nu = {\pm c_{1}}}}},{i.e.\mspace{14mu} {that}}} & ({A18}) \\{{\left. \frac{d\; \phi}{d\; \nu} \right|_{\nu = {\pm c_{1}}} = {\left. \left( \frac{{\pm 2}\; c_{1}}{\frac{\partial^{2}g}{\partial\phi^{2}}} \right)^{\frac{1}{2}} \right|_{\phi = \phi_{\mp}} = \frac{\left( {2\; c_{1}{\overset{.}{R}}_{\mp}} \right)^{\frac{1}{2}}}{\Delta^{\frac{1}{2}}}}},} & ({A19})\end{matrix}$

in which we have calculated (∂² _(g)/∂φ²) from (7) and (8). Theright-hand side of (A19) is, in turn, indeterminate on the cusp curve ofthe bifurcation surface (the envelope) where c₁=Δ=0. Removing thisindeterminacy by expanding the numerator in this expression in powers of

$\Delta^{\frac{1}{4}},$

we find that dφ/dυ assumes the value

$2^{\frac{1}{3}}$

at the cusp curve.

Hence, the coefficients p₀ and q₀ that appear in the expressions (A8)and (A9) for G₀ are explicitly given by

$\begin{matrix}{{p_{0} = {\left( {\omega/c} \right)\left( {\frac{1}{2}c_{1}} \right)^{\frac{1}{2}}\left( {{\hat{R}}_{-}^{- \frac{1}{2}} + {\hat{R}}_{+}^{- \frac{1}{2}}} \right)\Delta^{- \frac{1}{4}}}}{and}} & \left. ({A20}) \right) \\{q_{0} = {\left( {\omega/c} \right)\left( {2\; c_{1}} \right)^{- \frac{1}{2}}\left( {{\hat{R}}_{-}^{- \frac{1}{2}} - {\hat{R}}_{+}^{- \frac{1}{2}}} \right)\Delta^{- \frac{1}{4}}}} & ({A21})\end{matrix}$

[see (A4)-(A6) and (A19)].

In the regime of validity of (A8) and (A9), where Δ is much smaller than

$\left( {{{\hat{r}}_{p}^{2}{\hat{r}}^{2}} - \overset{\Cup}{1}} \right)^{\frac{1}{2}},$

the leading terms in the expressions for {circumflex over (R)}_(±), c₁,p₀ and q₀ are

$\begin{matrix}{{{\hat{R}}_{\pm} = {{\left( {{{\hat{r}}_{p}^{2}{\hat{r}}^{2}} - 1} \right)^{\frac{1}{2}} \pm {\left( {{{\hat{r}}_{p}^{2}{\hat{r}}^{2}} - 1} \right)^{- \frac{1}{2}}\Delta^{\frac{1}{2}}}} + {O(\Delta)}}},} & ({A22}) \\{{c_{1} = {{2^{- \frac{1}{3}}\left( {{{\hat{r}}_{p}^{2}{\hat{r}}^{2}} - 1} \right)^{- \frac{1}{2}}\Delta^{\frac{1}{2}}} + {O(\Delta)}}},} & ({A23}) \\{{p_{0} = {{2^{\frac{1}{3}}\left( {w/c} \right)\left( {{{\hat{r}}^{2}{\hat{r}}_{p}^{2}} - 1} \right)^{- \frac{1}{2}}} + {O\left( \Delta^{\frac{1}{2}} \right)}}},{and}} & \left( {A\; 24} \right) \\{q_{0} = {{2^{- \frac{1}{3}}\left( {w/c} \right)\left( {{{\hat{r}}^{2}{\hat{r}}_{p}^{2}} - 1} \right)^{- 1}} + {{O\left( \Delta^{\frac{1}{2}} \right)}.}}} & ({A25})\end{matrix}$

These may be obtained by using (9) to express {circumflex over (z)}everywhere in (10), (11) and (A2) in terms of Δ and {circumflex over(r)}, and expanding the resulting expressions in powers of

$\Delta^{\frac{1}{2}}.$

The quantity Δ in turn has the following value at points

${\hat{0} \leq {{\hat{z}}_{c} - \hat{z}}{\left( {{\hat{r}}_{p}^{2} - 1} \right)^{\frac{1}{2}}\left( {{\hat{r}}^{2} - 1} \right)^{\frac{1}{2}}}}:$

$\begin{matrix}{{\Delta = {{2\left( {{\hat{r}}_{p}^{2} - 1} \right)^{\frac{1}{2}}\left( {{\hat{r}}^{2} - 1} \right)^{\frac{1}{2}}\left( {{\hat{z}}_{c} - \hat{z}} \right)} + {O\left\lbrack \left( {{\hat{z}}_{c} - \hat{z}} \right)^{2} \right\rbrack}}},} & ({A26})\end{matrix}$

in which {circumflex over (z)}_(c) is given by the expression with theplus sign in (12b).

For an observation point in the far zone ({circumflex over(r)}_(P)>>1)the above expressions reduce to

$\begin{matrix}{{{\hat{R}}_{\pm} \simeq {\hat{r}\hat{r}p}},{c_{1} \simeq {2^{\frac{1}{6}}\left( {\hat{r}\hat{r}p} \right)^{- \frac{1}{2}}\left( {1 - {\hat{r}}^{- 2}} \right)^{\frac{1}{4}}\left( {{\hat{z}}_{c} - \hat{z}} \right)^{2}}},} & ({A27}) \\{{\Delta \; \simeq {2\hat{r}{p\left( {{\hat{r}}^{2} - 1} \right)}^{\frac{1}{2}}\left( {{\hat{z}}_{c} - \hat{z}} \right)}},} & ({A28}) \\{{p_{0} \simeq {2^{\frac{1}{3}}\left( {\omega/c} \right)\left( {\hat{r}\; p\; \hat{r}} \right)^{- 1}}},{q_{0} \simeq {2^{- \frac{1}{3}}\left( {\omega/c} \right)\left( {\hat{r}p\hat{r}} \right)^{- 2}}},{and}} & ({A29}) \\{{X \simeq {3\left( {\frac{1}{2}\hat{r}\hat{r}\; p} \right)^{\frac{3}{2}}\left( {1 - {\hat{r}}^{- 2}} \right)^{- \frac{3}{4}}{\left( {\varphi - \varphi_{c}} \right)/\left( {{\overset{\sim}{z}}_{c} - \overset{\sim}{z}} \right)^{\frac{3}{2}}}}},} & ({A30})\end{matrix}$

in which {circumflex over (z)}_(c)−{circumflex over (z)} has beenassumed to be finite.

Evaluation of the other Green's functions, G₁, G₂ and G₃, entailscalculations which have many steps in common with that of G₀. Since theintegrals in (34), (42) and (52) differ from that in (16) only in thattheir integrands respectively contain the extra factors {circumflex over(n)}, ê_(φ) and {circumflex over (n)}×ê_(φ), they can be rewritten asintegrals of the form (A3) in which the functions

f₁(v)≡{circumflex over (n)}f₀, f₂(v)≡ê_(φ)f₀ and f₃(v)≡{circumflex over(n)}×ê_(φ)f₀   (A31)

replace the f₀(v) given by (A4).

If p₀ and q₀ are correspondingly replaced, in accordance with (A5) and(A6), by

$\begin{matrix}{{p_{k} = {\frac{1}{2}\left( {f_{k}_{{v} = c_{1}}{{+ f_{k}}_{v = {- c_{1}}}}} \right)}},{k = 1},2,3,} & ({A32}) \\{{q_{k} = {\frac{1}{2}{c_{1}^{- 1}\left( {f_{k}_{v = c_{1}}{{- f_{k}}_{v = {- c_{1}}}}} \right)}}},{k = 1},2,3} & ({A33})\end{matrix}$

then every step of the analysis that led from (A7) to (A8) and (A9)would be equally applicable to the evaluation of G_(k). It follows,therefore, that

$\begin{matrix}{{{\left. G_{k}^{in} \right.\sim 2}{{c_{1}^{- 2}\left( {1 - x^{2}} \right)}^{- \frac{1}{2}}\left\lbrack {{p_{k}{\cos \left( {\frac{1}{3}\arcsin_{x}} \right)}} - {c_{1}q_{k}{\sin \left( {\frac{2}{3}\arcsin_{x}} \right)}}} \right\rbrack}},{{x} < 1},} & ({A34}) \\{G_{k}^{out} \sim {{c_{1}^{- 2}\left( {x^{2} - 1} \right)}^{- \frac{1}{2}}\left\lbrack {{{p_{k}{\sinh \left( {\frac{1}{3}{arc}\; \cosh {x}} \right)}} + \left. \quad{c_{1}q_{k}{{sgn}(x)}{\sinh \left( {\frac{2}{3}{arc}\; \cosh {x}} \right)}} \right\rbrack},{{x} > 1},} \right.}} & ({A35})\end{matrix}$

constitute the uniform asymptotic approximations to the functions G_(k)inside and outside the bifurcation surface (the envelope) |x|=1.

Explicit expressions for p_(k) and q_(k) as functions of (r, z) may befound from (8), (A19), and (A31)-(A33) jointly. The result is

$\begin{matrix}{{\begin{matrix}p_{1} \\q_{1}\end{matrix} = {2^{- \frac{1}{2}}\left( {\omega/c} \right)c_{1}^{\pm \frac{1}{2}}\Delta^{- \frac{1}{4}}\left\{ {{\left\{ {{\left( {{\hat{r}}_{p} - {\hat{r}}_{p}^{- 1}} \right)\left( {{\hat{R}}_{-}^{- \frac{3}{2}} \pm {\hat{R}}_{+}^{- \frac{3}{2}}} \right)} - {{\hat{r}}_{p}^{- 1}{\Delta^{\frac{1}{2}}\left( {{\hat{R}}_{-}^{- \frac{3}{2}} \mp {\hat{R}}_{+}^{- \frac{3}{2}}} \right)}}} \right\rbrack {\hat{e}}_{rp}} + {{{\hat{r}}_{P}^{- 1}\left( {{\hat{R}}_{-}^{- \frac{1}{2}} \pm {\hat{R}}_{+}^{- \frac{1}{2}}} \right)}{\hat{e}}_{\phi \; p}} + {\left( {{\hat{z}}_{p} - \hat{z}} \right)\left( {{\hat{R}}_{-}^{- \frac{3}{2}} \pm {\hat{R}}_{+}^{- \frac{3}{2}}} \right){\hat{e}}_{z_{p}}}} \right\}}},} & ({A36}) \\{{\begin{matrix}p_{2} \\q_{2}\end{matrix} = {2^{- \frac{1}{2}}\left( {\omega/c} \right)\left( {\hat{r}\hat{r}p} \right)^{- 1}c_{1}^{\pm \frac{1}{2}}\Delta^{- \frac{1}{4}}\left\{ {{\left( {{\hat{R}}_{-}^{\frac{1}{2}} \pm {\hat{R}}_{+}^{\frac{1}{2}}} \right){\hat{e}}_{rp}} + {\left\lbrack {{{\hat{R}}_{-}^{- \frac{1}{2}} \pm {\hat{R}}_{+}^{- \frac{1}{2}}} + {\Delta^{\frac{1}{2}}\left( {{\hat{R}}_{-}^{- \frac{1}{2}} \mp {\hat{R}}_{+}^{- \frac{1}{2}}} \right)}} \right\rbrack {\hat{e}}_{\phi \; p}}} \right\}}},{and}} & ({A37}) \\{{\begin{matrix}p_{3} \\q_{3}\end{matrix} = {2^{- \frac{1}{2}}\left( {\omega/c} \right)\left( {\hat{r}\hat{r}p} \right)^{- 1}c_{1}^{\pm \frac{1}{2}}\Delta^{- \frac{1}{4}}\left\{ {{{- {\left( {{\hat{z}}_{p} - \hat{z}} \right)\left\lbrack {{{\hat{R}}_{-}^{- \frac{3}{2}} \pm {\hat{R}}_{+}^{- \frac{3}{2}}} + {\Delta^{\frac{1}{2}}\left( {{\hat{R}}_{-}^{- \frac{3}{2}} \mp {\hat{R}}_{+}^{- \frac{3}{2}}} \right)}} \right\rbrack}}{\hat{e}}_{rp}} + {\left( {{\hat{z}}_{p} - \hat{z}} \right)\left( {{\hat{R}}_{-}^{- \frac{1}{2}} \pm {\hat{R}}_{+}^{- \frac{1}{2}}} \right){\hat{e}}_{\phi_{p}}} + {{{\hat{r}}_{p}\left\lbrack {{\Delta^{\frac{1}{2}}\left( {{\hat{R}}_{-}^{- \frac{3}{2}} \mp {\hat{R}}_{+}^{- \frac{3}{2}}} \right)} - {\left( {{\hat{r}}^{2} - 1} \right)\left( {{\hat{R}}_{-}^{- \frac{3}{2}} \pm {\hat{R}}_{+}^{- \frac{3}{2}}} \right)}} \right\rbrack}{\hat{e}}_{zp}}} \right\}}},} & ({A38})\end{matrix}$

where use has been made of the factê_(φ=−)sin(φ−φ_(P))ê_(r p)+cos(φ−φ_(P))ê_(φt). Here, the expressionswith the upper signs yield the p_(k) and those with the lower signs theq_(k).

The asymptotic value of each G_(k) ^(out) is indeterminate on thebifurcation surface (the envelope). If we expand the numerator of (A35)in powers of its denominator and cancel out the common factor

$\left( {x^{2} - 1} \right)^{\frac{1}{2}}$

prior to evaluating the ratio in this equation, we obtain

G _(k) ^(out)|_(φ=φ±) =G _(k) ^(out)|_(X=±1)˜(p _(k)±2c ₁ q _(k))/(3c ₁²).   (A39)

This shows that G_(k) ^(out)|_(φ=φ−) and G_(k) ^(out)|_(φ=φ+) remaindifferent even in the limit where the surfaces φ=φ⁻ and φ=φ₊ coalesce.The coefficients q_(k) that specify the strengths of the discontinuities

$\begin{matrix}{G_{k}^{out}_{\varphi = \varphi_{+}}{{- G_{k}^{out}}_{\varphi = \varphi_{-}}{\sim {\frac{4}{3}{q_{k}/c_{1}}}}}} & ({A40})\end{matrix}$

reduce to

$\begin{matrix}{{q_{1} \simeq {\frac{3}{2^{\frac{1}{3}}}\left( {\omega/c} \right){\left( {\hat{r}\hat{r}p} \right)^{- 3}\left\lbrack {{\left( {1 - {\frac{2}{3}{\hat{r}}^{2}}} \right)\hat{r}p{\hat{e}}_{rp}} + {\left( {{\hat{z}}_{p} - \hat{z}} \right){\hat{e}}_{zp}}} \right\rbrack}}},} & ({A41}) \\{{q_{2} \simeq {2^{\frac{2}{3}}\left( {\omega/c} \right)\left( {\hat{r}\hat{r}p} \right)^{- 1}{\hat{e}}_{\phi \; p}}},{and}} & ({A42}) \\{q_{3} \simeq {{- 2^{\frac{2}{3}}}\left( {\omega/c} \right){\left( {\hat{r}\hat{r}p} \right)^{- 2}\left\lbrack {{\left( {{\hat{z}}_{p} - \hat{z}} \right){\hat{e}}_{rp}} - {\hat{r}p{\hat{e}}_{zp}}} \right\rbrack}}} & ({A43})\end{matrix}$

in the regime of validity of (A27) and (A28).

When

$0 \leq {{\hat{z}}_{c} - \hat{z}}{\left( {{\hat{r}}^{2} - 1} \right)^{\frac{1}{2}}\hat{r}p}$

the expressions (A41) and (A43) further reduce to

$\begin{matrix}{{q_{1} \simeq {\frac{3}{2^{\frac{1}{3}}}\left( {\omega/c} \right)\left( {\hat{r}\hat{r}p} \right)^{- 2}n_{1}}},{{{and}\mspace{14mu} q_{3}} \simeq {2^{\frac{2}{3}}\left( {\omega/c} \right)\left( {\hat{r}\hat{r}p} \right)^{- 1}n_{3}}}} & ({A44})\end{matrix}$

with

$\begin{matrix}{{n_{1} \equiv {{\left( {{\hat{r}}^{- 1} - {\frac{2}{3}\hat{r}}} \right){\hat{e}}_{rp}} - {\left( {1 - {\hat{r}}^{- 2}} \right)^{\frac{1}{2}}{\hat{e}}_{zp}\mspace{14mu} {and}\mspace{14mu} n_{3}}} \equiv {{\left( {1 - {\hat{r}}^{- 2}} \right)^{\frac{1}{2}}{\hat{e}}_{rp}} + {{\hat{r}}^{- 1}{\hat{e}}_{zp}}}},} & ({A45})\end{matrix}$

for in this case (12b)—with the adopted plus sign—can be used to replace

$\hat{z} - {\hat{z}p\mspace{14mu} {by}\mspace{14mu} \left( {{\hat{r}}^{2} - 1} \right)^{\frac{1}{2}}\hat{r}{p.}}$

Therefore, at least the following is claimed:
 1. An apparatus forgenerating electromagnetic radiation, comprising: a series of adjacentelectrode pairs arranged on a common dielectric substrate, theelectrodes of each electrode pair substantially aligned on oppositesides of the common dielectric substrate; and electrode drive circuitryconfigured to: energize a first electrode pair in the series ofelectrode pairs at an energizing time to produce a volume polarizationdistribution pattern within the common dielectric substrate; energize anext electrode pair in the series of electrode pairs at a nextenergizing time to produce a variation of the volume polarizationdistribution pattern within the common dielectric substrate, the centerof the next electrode pair located a distance from the center of theprevious electrode pair, wherein the next energizing time is a timeinterval after the previous energizing time, the time interval less thanthe time for light to travel the distance between the centers of theprevious and the next electrode pairs.
 2. The apparatus of claim 1,wherein the electrode drive circuitry is further configured torepetitively energize the next adjacent electrode pair for subsequentelectrode pairs in the series of adjacent electrode pairs to produce acontinuous time-varying volume polarization distribution pattern withinthe common dielectric substrate.
 3. The apparatus of claim 1, whereinthe series of electrode pairs is a series of adjacent electrode pairs.4. The apparatus of claim 1, wherein the series of electrode pairs forma ring of electrode pairs.
 5. The apparatus of claim 1, wherein theseries of electrode pairs form a rectilinear configuration of electrodepairs.
 6. The apparatus of claim 1, wherein the distance between thecenters of adjacent electrode pairs is constant.
 7. The apparatus ofclaim 6, wherein the time interval between energizing times is constantor increasing.
 8. The apparatus of claim 6, wherein the time intervalbetween energizing times is decreasing.
 9. The apparatus of claim 1,wherein the electrode pairs are energized for a period of time greaterthan the time interval.
 10. The apparatus of claim 1, wherein theelectrode pairs are energized at a modulated voltage level.
 11. Theapparatus of claim 1, wherein the electrode drive circuitry is furtherconfigured to: energize the first electrode pair at a new energizingtime; and repetitively energize the next electrode pair for subsequentelectrode pairs in the series of electrode pairs.
 12. The apparatus ofclaim 11, wherein the series of electrode pairs form a ring of electrodepairs, the last electrode pair located a distance from the firstelectrode pair.
 13. The apparatus of claim 12, wherein the ring ofelectrode pairs is continuous.
 14. The apparatus of claim 12, whereinthe new energizing time is a time interval after the energizing time ofthe last electrode pair of the series, the time interval less than thetime for light to travel the distance between the centers of the lastelectrode pair and the first electrode pair.
 15. The apparatus of claim1, wherein the volume polarization distribution pattern is a volumedistribution pattern of polarization current.
 16. The apparatus of claim1, wherein the volume polarization distribution pattern is a volumedistribution pattern of polarization charge.